Exercise 1.11 1. Factorise. a) \( a b+c b+a d+c d \) c) \( a k-b k-a m+b m \) e) \( b y-c y-b+c \) g) \( a-b-a c+b c \) i) \( 5 a+b(a+5)+b^{2} \) k) \( a^{2}-a(c-5)-5 c \) m) \( 3 x y^{2}+6 x y-y(y w+2 w) \) o) \( x^{4}+x^{3}+5 x+5 \)

Factor the expression by following steps: - step0: Factor: \(x^{4}+x^{3}+5x+5\) - step1: Rewrite the expression: \(x^{4}+5x+x^{3}+5\) - step2: Factor the expression: \(x\left(x^{3}+5\right)+x^{3}+5\) - step3: Factor the expression: \(\left(x+1\right)\left(x^{3}+5\right)\) Factor the expression \( a b+c b+a d+c d \). Factor the expression by following steps: - step0: Factor: \(ab+cb+ad+cd\) - step1: Rewrite the expression: \(ab+ad+cb+cd\) - step2: Factor the expression: \(a\left(b+d\right)+c\left(b+d\right)\) - step3: Factor the expression: \(\left(a+c\right)\left(b+d\right)\) Factor the expression \( a-b-a c+b c \). Factor the expression by following steps: - step0: Factor: \(a-b-ac+bc\) - step1: Rewrite the expression: \(-ac+a+bc-b\) - step2: Factor the expression: \(-a\left(c-1\right)+b\left(c-1\right)\) - step3: Factor the expression: \(\left(-a+b\right)\left(c-1\right)\) - step4: Factor the expression: \(-\left(a-b\right)\left(c-1\right)\) Factor the expression \( b y-c y-b+c \). Factor the expression by following steps: - step0: Factor: \(by-cy-b+c\) - step1: Rewrite the expression: \(by-b-cy+c\) - step2: Factor the expression: \(b\left(y-1\right)-c\left(y-1\right)\) - step3: Factor the expression: \(\left(b-c\right)\left(y-1\right)\) Factor the expression \( a k-b k-a m+b m \). Factor the expression by following steps: - step0: Factor: \(ak-bk-am+bm\) - step1: Rewrite the expression: \(ak-am-bk+bm\) - step2: Factor the expression: \(a\left(k-m\right)-b\left(k-m\right)\) - step3: Factor the expression: \(\left(a-b\right)\left(k-m\right)\) Factor the expression \( 5 a+b(a+5)+b^{2} \). Factor the expression by following steps: - step0: Factor: \(5a+b\left(a+5\right)+b^{2}\) - step1: Calculate: \(5a+b^{2}+b\left(a+5\right)\) - step2: Simplify: \(5a+b^{2}+ba+5b\) - step3: Rewrite the expression: \(b^{2}+5b+ab+5a\) - step4: Factor the expression: \(b\left(b+5\right)+a\left(b+5\right)\) - step5: Factor the expression: \(\left(b+a\right)\left(b+5\right)\) Factor the expression \( a^{2}-a(c-5)-5 c \). Factor the expression by following steps: - step0: Factor: \(a^{2}-a\left(c-5\right)-5c\) - step1: Calculate: \(a^{2}-5c-a\left(c-5\right)\) - step2: Simplify: \(a^{2}-5c-ac+5a\) - step3: Rewrite the expression: \(a^{2}+5a-ca-5c\) - step4: Factor the expression: \(a\left(a+5\right)-c\left(a+5\right)\) - step5: Factor the expression: \(\left(a-c\right)\left(a+5\right)\) Factor the expression \( 3 x y^{2}+6 x y-y(y w+2 w) \). Factor the expression by following steps: - step0: Factor: \(3xy^{2}+6xy-y\left(yw+2w\right)\) - step1: Factor the expression: \(\left(3x-w\right)\left(y^{2}+2y\right)\) - step2: Calculate: \(\left(3x-w\right)y\left(y+2\right)\) - step3: Calculate: \(y\left(3x-w\right)\left(y+2\right)\) Here are the factorizations for each expression: 1. **a)** \( ab + cb + ad + cd \) **Factored form:** \((a+c)(b+d)\) 2. **c)** \( ak - bk - am + bm \) **Factored form:** \((a-b)(k-m)\) 3. **e)** \( by - cy - b + c \) **Factored form:** \((b-c)(y-1)\) 4. **g)** \( a - b - ac + bc \) **Factored form:** \(-(a-b)(c-1)\) 5. **i)** \( 5a + b(a + 5) + b^2 \) **Factored form:** \((b+a)(b+5)\) 6. **k)** \( a^2 - a(c - 5) - 5c \) **Factored form:** \((a-c)(a+5)\) 7. **m)** \( 3xy^2 + 6xy - y(yw + 2w) \) **Factored form:** \(y(3x-w)(y+2)\) 8. **o)** \( x^4 + x^3 + 5x + 5 \) **Factored form:** \((x+1)(x^3+5)\) If you have any further questions or need additional assistance, feel free to ask!