EXERCISE 4 Fully simplify each of the follos 1. \( (2 x+y)\left(3 x^{2}+2 x y+5 y^{2}\right) \) 3. \( (x-y)(3 x-2 x y+4) \) 5. \( (7 m-3 n)\left(5 m^{2}-2 m n-7 n^{2}\right) \) 7. \( (2 x-y)\left(4 x^{2}+2 x y-y^{2}\right) \) 9. \( (4 m-n)\left(2 m^{2}-3 m n-n^{2}\right) \) 11. \( (2 x-y)(2 x+y)(3 x-5 x y-y \) 13. \( (9 m-n)\left(3 m^{2}-m n+2 n^{2}\right) \) Cket \( 15.5(a-b)\left(a^{2}+a b+b^{2}\right) \) 17. \( 2(3 p+q)\left(9 p^{2}-3 p q+q^{2}\right) \) 19. \( 3(2 x+1)\left(x^{2}-5 x-1\right) \)

Simplify the expression by following steps: - step0: Calculate: \(2\left(3p+q\right)\left(9p^{2}-3pq+q^{2}\right)\) - step1: Simplify the product: \(2\left(27p^{3}+q^{3}\right)\) - step2: Calculate: \(54p^{3}+2q^{3}\) Expand the expression \( 15.5(a-b)(a^{2}+a b+b^{2}) \) Simplify the expression by following steps: - step0: Calculate: \(15.5\left(a-b\right)\left(a^{2}+ab+b^{2}\right)\) - step1: Multiply the terms: \(\left(15.5a-15.5b\right)\left(a^{2}+ab+b^{2}\right)\) - step2: Factor the expression: \(\frac{31}{2}\left(a-b\right)\left(a^{2}+ab+b^{2}\right)\) - step3: Simplify the product: \(\frac{31}{2}\left(a^{3}-b^{3}\right)\) - step4: Calculate: \(\frac{31}{2}a^{3}-\frac{31}{2}b^{3}\) Expand the expression \( (x-y)(3 x-2 x y+4) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(x-y\right)\left(3x-2xy+4\right)\) - step1: Apply the distributive property: \(x\times 3x-x\times 2xy+x\times 4-y\times 3x-\left(-y\times 2xy\right)-y\times 4\) - step2: Multiply the terms: \(3x^{2}-2x^{2}y+4x-3yx-\left(-2y^{2}x\right)-4y\) - step3: Remove the parentheses: \(3x^{2}-2x^{2}y+4x-3yx+2y^{2}x-4y\) Expand the expression \( (4 m-n)(2 m^{2}-3 m n-n^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(4m-n\right)\left(2m^{2}-3mn-n^{2}\right)\) - step1: Apply the distributive property: \(4m\times 2m^{2}-4m\times 3mn-4mn^{2}-n\times 2m^{2}-\left(-n\times 3mn\right)-\left(-n\times n^{2}\right)\) - step2: Multiply the terms: \(8m^{3}-12m^{2}n-4mn^{2}-2nm^{2}-\left(-3n^{2}m\right)-\left(-n^{3}\right)\) - step3: Remove the parentheses: \(8m^{3}-12m^{2}n-4mn^{2}-2nm^{2}+3n^{2}m+n^{3}\) - step4: Subtract the terms: \(8m^{3}-14m^{2}n-mn^{2}+n^{3}\) Expand the expression \( (9 m-n)(3 m^{2}-m n+2 n^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(9m-n\right)\left(3m^{2}-mn+2n^{2}\right)\) - step1: Apply the distributive property: \(9m\times 3m^{2}-9m\times mn+9m\times 2n^{2}-n\times 3m^{2}-\left(-nmn\right)-n\times 2n^{2}\) - step2: Multiply the terms: \(27m^{3}-9m^{2}n+18mn^{2}-3nm^{2}-\left(-n^{2}m\right)-2n^{3}\) - step3: Remove the parentheses: \(27m^{3}-9m^{2}n+18mn^{2}-3nm^{2}+n^{2}m-2n^{3}\) - step4: Subtract the terms: \(27m^{3}-12m^{2}n+19mn^{2}-2n^{3}\) Expand the expression \( 3(2 x+1)(x^{2}-5 x-1) \) Simplify the expression by following steps: - step0: Calculate: \(3\left(2x+1\right)\left(x^{2}-5x-1\right)\) - step1: Multiply the terms: \(\left(6x+3\right)\left(x^{2}-5x-1\right)\) - step2: Apply the distributive property: \(6x\times x^{2}-6x\times 5x-6x\times 1+3x^{2}-3\times 5x-3\times 1\) - step3: Multiply the terms: \(6x^{3}-30x^{2}-6x+3x^{2}-15x-3\) - step4: Add the terms: \(6x^{3}-27x^{2}-21x-3\) Expand the expression \( (2 x-y)(2 x+y)(3 x-5 x y-y) \) Simplify the expression by following steps: - step0: Calculate: \(\left(2x-y\right)\left(2x+y\right)\left(3x-5xy-y\right)\) - step1: Multiply the terms: \(\left(4x^{2}-y^{2}\right)\left(3x-5xy-y\right)\) - step2: Apply the distributive property: \(4x^{2}\times 3x-4x^{2}\times 5xy-4x^{2}y-y^{2}\times 3x-\left(-y^{2}\times 5xy\right)-\left(-y^{2}\times y\right)\) - step3: Multiply the terms: \(12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x-\left(-5y^{3}x\right)-\left(-y^{3}\right)\) - step4: Remove the parentheses: \(12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x+5y^{3}x+y^{3}\) Expand the expression \( (7 m-3 n)(5 m^{2}-2 m n-7 n^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(7m-3n\right)\left(5m^{2}-2mn-7n^{2}\right)\) - step1: Apply the distributive property: \(7m\times 5m^{2}-7m\times 2mn-7m\times 7n^{2}-3n\times 5m^{2}-\left(-3n\times 2mn\right)-\left(-3n\times 7n^{2}\right)\) - step2: Multiply the terms: \(35m^{3}-14m^{2}n-49mn^{2}-15nm^{2}-\left(-6n^{2}m\right)-\left(-21n^{3}\right)\) - step3: Remove the parentheses: \(35m^{3}-14m^{2}n-49mn^{2}-15nm^{2}+6n^{2}m+21n^{3}\) - step4: Subtract the terms: \(35m^{3}-29m^{2}n-43mn^{2}+21n^{3}\) Expand the expression \( (2 x-y)(4 x^{2}+2 x y-y^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2x-y\right)\left(4x^{2}+2xy-y^{2}\right)\) - step1: Apply the distributive property: \(2x\times 4x^{2}+2x\times 2xy-2xy^{2}-y\times 4x^{2}-y\times 2xy-\left(-y\times y^{2}\right)\) - step2: Multiply the terms: \(8x^{3}+4x^{2}y-2xy^{2}-4yx^{2}-2y^{2}x-\left(-y^{3}\right)\) - step3: Remove the parentheses: \(8x^{3}+4x^{2}y-2xy^{2}-4yx^{2}-2y^{2}x+y^{3}\) - step4: Subtract the terms: \(8x^{3}+0-2xy^{2}-2y^{2}x+y^{3}\) - step5: Remove 0: \(8x^{3}-2xy^{2}-2y^{2}x+y^{3}\) - step6: Subtract the terms: \(8x^{3}-4xy^{2}+y^{3}\) Expand the expression \( (2 x+y)(3 x^{2}+2 x y+5 y^{2}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2x+y\right)\left(3x^{2}+2xy+5y^{2}\right)\) - step1: Apply the distributive property: \(2x\times 3x^{2}+2x\times 2xy+2x\times 5y^{2}+y\times 3x^{2}+y\times 2xy+y\times 5y^{2}\) - step2: Multiply the terms: \(6x^{3}+4x^{2}y+10xy^{2}+3yx^{2}+2y^{2}x+5y^{3}\) - step3: Add the terms: \(6x^{3}+7x^{2}y+12xy^{2}+5y^{3}\) Here are the fully simplified results for each of the expressions: 1. **Expression:** \( (2 x+y)(3 x^{2}+2 x y+5 y^{2}) \) **Simplified Result:** \( 6x^{3}+7x^{2}y+12xy^{2}+5y^{3} \) 2. **Expression:** \( (x-y)(3 x-2 x y+4) \) **Simplified Result:** \( 3x^{2}-2x^{2}y+4x-3yx+2y^{2}x-4y \) 3. **Expression:** \( (7 m-3 n)(5 m^{2}-2 m n-7 n^{2}) \) **Simplified Result:** \( 35m^{3}-29m^{2}n-43mn^{2}+21n^{3} \) 4. **Expression:** \( (2 x-y)(4 x^{2}+2 x y-y^{2}) \) **Simplified Result:** \( 8x^{3}-4xy^{2}+y^{3} \) 5. **Expression:** \( (4 m-n)(2 m^{2}-3 m n-n^{2}) \) **Simplified Result:** \( 8m^{3}-14m^{2}n-mn^{2}+n^{3} \) 6. **Expression:** \( (2 x-y)(2 x+y)(3 x-5 x y-y) \) **Simplified Result:** \( 12x^{3}-20x^{3}y-4x^{2}y-3y^{2}x+5y^{3}x+y^{3} \) 7. **Expression:** \( (9 m-n)(3 m^{2}-m n+2 n^{2}) \) **Simplified Result:** \( 27m^{3}-12m^{2}n+19mn^{2}-2n^{3} \) 8. **Expression:** \( 15.5(a-b)(a^{2}+a b+b^{2}) \) **Simplified Result:** \( \frac{31}{2}a^{3}-\frac{31}{2}b^{3} \) 9. **Expression:** \( 2(3 p+q)(9 p^{2}-3 p q+q^{2}) \) **Simplified Result:** \( 54p^{3}+2q^{3} \) 10. **Expression:** \( 3(2 x+1)(x^{2}-5 x-1) \) **Simplified Result:** \( 6x^{3}-27x^{2}-21x-3 \) If you need further assistance or explanations for any specific expression, feel free to ask!