3) \( \frac{4}{a-2}+\frac{2 a-3}{a^{2}+2 a+4}-\frac{2 a^{2}-4 a+5}{a^{3}-8} \) 4) \( \frac{1}{a^{2}+3 a+2}+\frac{1}{a^{2}+5 a+6} \) 5) \( \frac{2}{a^{2}-4 a+3}-\frac{2}{a^{2}-8 a+15} \) 6) \( \frac{a^{2}-(b-c)^{2}}{(a+c)^{2}-b^{2}}+\frac{b^{2}-(a-c)^{2}}{(a+b)^{2}-c^{2}}+\frac{c^{2}-(a-b)^{2}}{(b+c)^{2}-a^{2}} \) 7) \( \frac{1}{(a-b)(a-c)}+\frac{1}{(b-a)(b-c)}+\frac{1}{(c-a)(c-b)} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{1}{a^{2}+3a+2}+\frac{1}{a^{2}+5a+6}\) - step1: Factor the expression: \(\frac{1}{\left(a+1\right)\left(a+2\right)}+\frac{1}{\left(a+3\right)\left(a+2\right)}\) - step2: Reduce fractions to a common denominator: \(\frac{a+3}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}+\frac{a+1}{\left(a+3\right)\left(a+2\right)\left(a+1\right)}\) - step3: Rewrite the expression: \(\frac{a+3}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}+\frac{a+1}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}\) - step4: Transform the expression: \(\frac{a+3+a+1}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}\) - step5: Add the terms: \(\frac{2a+4}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}\) - step6: Factor the expression: \(\frac{2\left(a+2\right)}{\left(a+1\right)\left(a+2\right)\left(a+3\right)}\) - step7: Reduce the fraction: \(\frac{2}{\left(a+1\right)\left(a+3\right)}\) - step8: Multiply the terms: \(\frac{2}{a^{2}+4a+3}\) Calculate or simplify the expression \( \frac{2}{a^{2}-4 a+3}-\frac{2}{a^{2}-8 a+15} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{a^{2}-4a+3}-\frac{2}{a^{2}-8a+15}\) - step1: Factor the expression: \(\frac{2}{\left(a-1\right)\left(a-3\right)}-\frac{2}{\left(a-5\right)\left(a-3\right)}\) - step2: Reduce fractions to a common denominator: \(\frac{2\left(a-5\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}-\frac{2\left(a-1\right)}{\left(a-5\right)\left(a-3\right)\left(a-1\right)}\) - step3: Rewrite the expression: \(\frac{2\left(a-5\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}-\frac{2\left(a-1\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step4: Transform the expression: \(\frac{2\left(a-5\right)-2\left(a-1\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step5: Multiply the terms: \(\frac{2a-10-2\left(a-1\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step6: Multiply the terms: \(\frac{2a-10-\left(2a-2\right)}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step7: Subtract the terms: \(\frac{-8}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step8: Rewrite the fraction: \(-\frac{8}{\left(a-1\right)\left(a-3\right)\left(a-5\right)}\) - step9: Calculate: \(-\frac{8}{a^{3}-9a^{2}+23a-15}\) Calculate or simplify the expression \( \frac{4}{a-2}+\frac{2 a-3}{a^{2}+2 a+4}-\frac{2 a^{2}-4 a+5}{a^{3}-8} \). Simplify the expression by following steps: - step0: Solution: \(\frac{4}{a-2}+\frac{2a-3}{a^{2}+2a+4}-\frac{2a^{2}-4a+5}{a^{3}-8}\) - step1: Factor the expression: \(\frac{4}{a-2}+\frac{2a-3}{a^{2}+2a+4}-\frac{2a^{2}-4a+5}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step2: Reduce fractions to a common denominator: \(\frac{4\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}+\frac{\left(2a-3\right)\left(a-2\right)}{\left(a^{2}+2a+4\right)\left(a-2\right)}-\frac{2a^{2}-4a+5}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step3: Rewrite the expression: \(\frac{4\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}+\frac{\left(2a-3\right)\left(a-2\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2a^{2}-4a+5}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step4: Transform the expression: \(\frac{4\left(a^{2}+2a+4\right)+\left(2a-3\right)\left(a-2\right)-\left(2a^{2}-4a+5\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step5: Multiply the terms: \(\frac{4a^{2}+8a+16+\left(2a-3\right)\left(a-2\right)-\left(2a^{2}-4a+5\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step6: Multiply the terms: \(\frac{4a^{2}+8a+16+2a^{2}-7a+6-\left(2a^{2}-4a+5\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step7: Calculate: \(\frac{4a^{2}+5a+17}{\left(a-2\right)\left(a^{2}+2a+4\right)}\) - step8: Simplify the product: \(\frac{4a^{2}+5a+17}{a^{3}-8}\) Calculate or simplify the expression \( \frac{1}{(a-b)(a-c)}+\frac{1}{(b-a)(b-c)}+\frac{1}{(c-a)(c-b)} \). Simplify the expression by following steps: - step0: Solution: \(\frac{1}{\left(a-b\right)\left(a-c\right)}+\frac{1}{\left(b-a\right)\left(b-c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}\) - step1: Rewrite the fractions: \(\frac{1}{\left(a-b\right)\left(a-c\right)}+\frac{1}{\left(b-a\right)\left(b-c\right)}-\frac{1}{\left(-c+a\right)\left(c-b\right)}\) - step2: Reduce fractions to a common denominator: \(\frac{-b+c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{a-c}{\left(b-a\right)\left(b-c\right)\left(a-c\right)}-\frac{a-b}{\left(-c+a\right)\left(c-b\right)\left(a-b\right)}\) - step3: Rewrite the expression: \(\frac{-b+c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{a-c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}-\frac{a-b}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}\) - step4: Transform the expression: \(\frac{-b+c+a-c-\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}\) - step5: Calculate: \(\frac{0}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}\) - step6: Calculate: \(0\) Calculate or simplify the expression \( \frac{a^{2}-(b-c)^{2}}{(a+c)^{2}-b^{2}}+\frac{b^{2}-(a-c)^{2}}{(a+b)^{2}-c^{2}}+\frac{c^{2}-(a-b)^{2}}{(b+c)^{2}-a^{2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{a^{2}-\left(b-c\right)^{2}}{\left(a+c\right)^{2}-b^{2}}+\frac{b^{2}-\left(a-c\right)^{2}}{\left(a+b\right)^{2}-c^{2}}+\frac{c^{2}-\left(a-b\right)^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step1: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{\left(a+c\right)^{2}-b^{2}}+\frac{b^{2}-\left(a-c\right)^{2}}{\left(a+b\right)^{2}-c^{2}}+\frac{c^{2}-\left(a-b\right)^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step2: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{a^{2}+2ac+c^{2}-b^{2}}+\frac{b^{2}-\left(a-c\right)^{2}}{\left(a+b\right)^{2}-c^{2}}+\frac{c^{2}-\left(a-b\right)^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step3: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{a^{2}+2ac+c^{2}-b^{2}}+\frac{b^{2}-a^{2}+2ac-c^{2}}{\left(a+b\right)^{2}-c^{2}}+\frac{c^{2}-\left(a-b\right)^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step4: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{a^{2}+2ac+c^{2}-b^{2}}+\frac{b^{2}-a^{2}+2ac-c^{2}}{a^{2}+2ab+b^{2}-c^{2}}+\frac{c^{2}-\left(a-b\right)^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step5: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{a^{2}+2ac+c^{2}-b^{2}}+\frac{b^{2}-a^{2}+2ac-c^{2}}{a^{2}+2ab+b^{2}-c^{2}}+\frac{c^{2}-a^{2}+2ab-b^{2}}{\left(b+c\right)^{2}-a^{2}}\) - step6: Expand the expression: \(\frac{a^{2}-b^{2}+2bc-c^{2}}{a^{2}+2ac+c^{2}-b^{2}}+\frac{b^{2}-a^{2}+2ac-c^{2}}{a^{2}+2ab+b^{2}-c^{2}}+\frac{c^{2}-a^{2}+2ab-b^{2}}{b^{2}+2bc+c^{2}-a^{2}}\) - step7: Divide the terms: \(\frac{a+b-c}{a+c+b}+\frac{b^{2}-a^{2}+2ac-c^{2}}{a^{2}+2ab+b^{2}-c^{2}}+\frac{c^{2}-a^{2}+2ab-b^{2}}{b^{2}+2bc+c^{2}-a^{2}}\) - step8: Divide the terms: \(\frac{a+b-c}{a+c+b}+\frac{b-a+c}{a+b+c}+\frac{c^{2}-a^{2}+2ab-b^{2}}{b^{2}+2bc+c^{2}-a^{2}}\) - step9: Divide the terms: \(\frac{a+b-c}{a+c+b}+\frac{b-a+c}{a+b+c}+\frac{c+a-b}{b+c+a}\) - step10: Rewrite the expression: \(\frac{a+b-c}{a+c+b}+\frac{b-a+c}{a+c+b}+\frac{c+a-b}{a+c+b}\) - step11: Transform the expression: \(\frac{a+b-c+b-a+c+c+a-b}{a+c+b}\) - step12: Calculate: \(\frac{a+b+c}{a+c+b}\) - step13: Reduce the fraction: \(1\) Keling, har bir ifodani alohida ko'rib chiqamiz va natijalarni taqdim etamiz: ### 3) \[ \frac{4}{a-2}+\frac{2 a-3}{a^{2}+2 a+4}-\frac{2 a^{2}-4 a+5}{a^{3}-8} \] Natija: \[ \frac{4a^{2}+5a+17}{a^{3}-8} \] ### 4) \[ \frac{1}{a^{2}+3 a+2}+\frac{1}{a^{2}+5 a+6} \] Natija: \[ \frac{2}{a^{2}+4a+3} \] ### 5) \[ \frac{2}{a^{2}-4 a+3}-\frac{2}{a^{2}-8 a+15} \] Natija: \[ -\frac{8}{a^{3}-9a^{2}+23a-15} \] ### 6) \[ \frac{a^{2}-(b-c)^{2}}{(a+c)^{2}-b^{2}}+\frac{b^{2}-(a-c)^{2}}{(a+b)^{2}-c^{2}}+\frac{c^{2}-(a-b)^{2}}{(b+c)^{2}-a^{2}} \] Natija: \[ 1 \] ### 7) \[ \frac{1}{(a-b)(a-c)}+\frac{1}{(b-a)(b-c)}+\frac{1}{(c-a)(c-b)} \] Natija: \[ 0 \] Agar sizda qo'shimcha savollar bo'lsa yoki boshqa masalalar bo'yicha yordam kerak bo'lsa, iltimos, xabar bering!