Factorise completely: 1. \( t x-t y+r x-r y \) 2. \( b y-a x+b x-a y \) 3. \( 3 a^{2}-a x+6 a b-2 b x \) 4. \( x^{3}+3 x^{2}-x-3 \) 5. \( 2 a-1-2 a b+b \) 6. \( 6 x^{2}-a y-2 a x+3 x y \) 7. \( 4 a^{2}-4 b^{2}-b^{3}+a^{2} b \) 8. \( 6 x^{2}-x^{5}-6+x^{3} \) 9. \( x^{4}-x^{2}-x^{2} y^{2}+y^{2} \) 10. \( x^{3}+y^{3}+x+y \)

Let's factorize each expression step by step: --- **1. \( t x - t y + r x - r y \)** Factor by grouping: \[ \begin{align*} t x - t y + r x - r y &= t(x - y) + r(x - y) \\ &= (x - y)(t + r) \end{align*} \] **Factorized Form:** \((t + r)(x - y)\) --- **2. \( b y - a x + b x - a y \)** Rearrange and factor by grouping: \[ \begin{align*} b y - a y + b x - a x &= y(b - a) + x(b - a) \\ &= (b - a)(x + y) \end{align*} \] **Factorized Form:** \((b - a)(x + y)\) --- **3. \( 3 a^{2} - a x + 6 a b - 2 b x \)** Factor by grouping: \[ \begin{align*} 3 a^{2} - a x + 6 a b - 2 b x &= 3a(a + 2b) - x(a + 2b) \\ &= (3a - x)(a + 2b) \end{align*} \end{align*} \] **Factorized Form:** \((3a - x)(a + 2b)\) --- **4. \( x^{3} + 3 x^{2} - x - 3 \)** Factor by grouping: \[ \begin{align*} x^{3} + 3 x^{2} - x - 3 &= x^{2}(x + 3) -1(x + 3) \\ &= (x^{2} - 1)(x + 3) \\ &= (x - 1)(x + 1)(x + 3) \end{align*} \end{align*} \] **Factorized Form:** \((x - 1)(x + 1)(x + 3)\) --- **5. \( 2 a -1 -2 a b + b \)** Rearrange and factor: \[ \begin{align*} 2 a -1 -2 a b + b &= 2a(1 - b) + (b - 1) \\ &= (1 - b)(2a - 1) \end{align*} \end{align*} \] **Factorized Form:** \((2a - 1)(1 - b)\) --- **6. \( 6 x^{2} -a y -2 a x + 3 x y \)** Factor by grouping: \[ \begin{align*} 6x^{2} -2a x + 3x y -a y &= 2x(3x - a) + y(3x - a) \\ &= (3x - a)(2x + y) \end{align*} \end{align*} \] **Factorized Form:** \((3x - a)(2x + y)\) --- **7. \( 4 a^{2} - 4 b^{2} - b^{3} + a^{2} b \)** Factor by grouping: \[ \begin{align*} 4 a^{2} -4 b^{2} - b^{3} +a^{2} b &= a^{2}(4 + b) - b^{2}(4 + b) \\ &= (a^{2} - b^{2})(4 + b) \\ &= (a - b)(a + b)(4 + b) \end{align*} \end{align*} \] **Factorized Form:** \((a - b)(a + b)(4 + b)\) --- **8. \( 6 x^{2} -x^{5} -6 +x^{3} \)** Rearrange and factor: \[ \begin{align*} 6x^{2} -x^{5} -6 +x^{3} &= -x^{5} + x^{3} + 6x^{2} - 6 \\ &= -\left(x^{5} - x^{3} - 6x^{2} + 6\right) \\ &= -\left(x^{3}(x^{2} - 1) -6(x^{2} -1)\right) \\ &= - (x^{3} - 6)(x^{2} -1) \\ &= - (x^{3} - 6)(x - 1)(x + 1) \end{align*} \end{align*} \] **Factorized Form:** \(- (x^{3} - 6)(x - 1)(x + 1)\) --- **9. \( x^{4} - x^{2} - x^{2} y^{2} + y^{2} \)** Rearrange and factor by grouping: \[ \begin{align*} x^{4} -x^{2} y^{2} -x^{2} + y^{2} &= x^{2}(x^{2} - y^{2}) -1(x^{2} - y^{2}) \\ &= (x^{2} -1)(x^{2} - y^{2}) \\ &= (x -1)(x +1)(x - y)(x + y) \end{align*} \end{align*} \] **Factorized Form:** \((x -1)(x +1)(x - y)(x + y)\) --- **10. \( x^{3} + y^{3} + x + y \)** Factor by grouping: \[ \begin{align*} x^{3} + y^{3} +x + y &= (x^{3} + y^{3}) + (x + y) \\ &= (x + y)(x^{2} -xy + y^{2}) + (x + y) \\ &= (x + y)(x^{2} -xy + y^{2} + 1) \end{align*} \end{align*} \] **Factorized Form:** \((x + y)(x^{2} - xy + y^{2} + 1)\) --- Each expression has been fully factorized.