Exorcifo 1.14 You may have to rearrange and group the terms. Check some of your answers by multiplying the factors. Try to find more than one way of grouping for some of the questions. Factorise. \( \begin{array}{ll}\text { a) } a^{2}-b^{2}+a-b & \text { b) } p^{2}+p-q^{2}-q \\ \text { c) } 4 x^{2}+2 x-9 y^{2}+3 y & \text { d) } 5 a^{2} b^{2}-4 b^{4}+25 a^{4} b^{4}-2 b^{2} \\ \text { e) } 3 a^{2}-2 b-3 b^{2}+2 a & \text { f) } x^{3}-4 x y^{2}+x y-2 y^{2}\end{array} \)

Factor the expression by following steps: - step0: Factor: \(a^{2}-b^{2}+a-b\) - step1: Rewrite the expression: \(a^{2}-ab+ba-b^{2}+a-b\) - step2: Factor the expression: \(a\left(a-b\right)+b\left(a-b\right)+a-b\) - step3: Factor the expression: \(\left(a+b+1\right)\left(a-b\right)\) Factor the expression \( 4 x^{2}+2 x-9 y^{2}+3 y \). Factor the expression by following steps: - step0: Factor: \(4x^{2}+2x-9y^{2}+3y\) - step1: Rewrite the expression: \(4x^{2}-6xy+2x+6yx-9y^{2}+3y\) - step2: Factor the expression: \(2x\left(2x-3y+1\right)+3y\left(2x-3y+1\right)\) - step3: Factor the expression: \(\left(2x+3y\right)\left(2x-3y+1\right)\) Factor the expression \( 3 a^{2}-2 b-3 b^{2}+2 a \). Factor the expression by following steps: - step0: Factor: \(3a^{2}-2b-3b^{2}+2a\) - step1: Rewrite the expression: \(3a^{2}-3ab+3ba-3b^{2}+2a-2b\) - step2: Factor the expression: \(3a\left(a-b\right)+3b\left(a-b\right)+2\left(a-b\right)\) - step3: Factor the expression: \(\left(3a+3b+2\right)\left(a-b\right)\) Factor the expression \( x^{3}-4 x y^{2}+x y-2 y^{2 \). Factor the expression by following steps: - step0: Factor: \(x^{3}-4xy^{2}+xy-2y^{2}\) - step1: Rewrite the expression: \(x^{3}-2x^{2}y+2x^{2}y-4xy^{2}+yx-2y^{2}\) - step2: Factor the expression: \(x^{2}\left(x-2y\right)+2xy\left(x-2y\right)+y\left(x-2y\right)\) - step3: Factor the expression: \(\left(x^{2}+2xy+y\right)\left(x-2y\right)\) Factor the expression \( p^{2}+p-q^{2}-q \). Factor the expression by following steps: - step0: Factor: \(p^{2}+p-q^{2}-q\) - step1: Rewrite the expression: \(p^{2}-pq+qp-q^{2}+p-q\) - step2: Factor the expression: \(p\left(p-q\right)+q\left(p-q\right)+p-q\) - step3: Factor the expression: \(\left(p+q+1\right)\left(p-q\right)\) Factor the expression \( 5 a^{2} b^{2}-4 b^{4}+25 a^{4} b^{4}-2 b^{2 \). Factor the expression by following steps: - step0: Factor: \(5a^{2}b^{2}-4b^{4}+25a^{4}b^{4}-2b^{2}\) - step1: Evaluate: \(25b^{4}a^{4}-4b^{4}+5b^{2}a^{2}-2b^{2}\) - step2: Factor the expression: \(b^{2}\left(25a^{4}b^{2}-4b^{2}+5a^{2}-2\right)\) - step3: Factor the expression: \(b^{2}\left(5a^{2}b^{2}+2b^{2}+1\right)\left(5a^{2}-2\right)\) Here are the factorizations for each of the expressions provided: ### a) \( a^{2}-b^{2}+a-b \) To factor this expression, we can rearrange and group the terms: \[ a^{2} + a - b^{2} - b = (a + b + 1)(a - b) \] **Factorization:** \((a + b + 1)(a - b)\) ### b) \( p^{2}+p-q^{2}-q \) Rearranging and grouping gives us: \[ p^{2} + p - q^{2} - q = (p + q + 1)(p - q) \] **Factorization:** \((p + q + 1)(p - q)\) ### c) \( 4x^{2}+2x-9y^{2}+3y \) Rearranging and grouping the terms: \[ 4x^{2} + 2x - 9y^{2} + 3y = (2x + 3y)(2x - 3y + 1) \] **Factorization:** \((2x + 3y)(2x - 3y + 1)\) ### d) \( 5a^{2}b^{2}-4b^{4}+25a^{4}b^{4}-2b^{2} \) This expression can be factored as: \[ 5a^{2}b^{2} - 4b^{4} + 25a^{4}b^{4} - 2b^{2} = b^{2}(5a^{2}b^{2} + 2b^{2} + 1)(5a^{2} - 2) \] **Factorization:** \(b^{2}(5a^{2}b^{2} + 2b^{2} + 1)(5a^{2} - 2)\) ### e) \( 3a^{2}-2b-3b^{2}+2a \) Rearranging and grouping gives: \[ 3a^{2} + 2a - 3b^{2} - 2b = (3a + 3b + 2)(a - b) \] **Factorization:** \((3a + 3b + 2)(a - b)\) ### f) \( x^{3}-4xy^{2}+xy-2y^{2} \) Rearranging and grouping gives: \[ x^{3} + xy - 4xy^{2} - 2y^{2} = (x^{2} + 2xy + y)(x - 2y) \] **Factorization:** \((x^{2} + 2xy + y)(x - 2y)\) These factorizations can be verified by multiplying the factors back together to ensure they yield the original expressions.