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. \( x^{3}+y^{3}+x+y \)

Here are the complete factorizations for each of the given expressions: 1. **\( t x - t y + r x - r y \)** Factor by grouping: \[ t(x - y) + r(x - y) = (t + r)(x - y) \] **Factorized Form:** \((t + r)(x - y)\) 2. **\( b y - a x + b x - a y \)** Rearrange and factor: \[ b(x + y) - a(x + y) = (b - a)(x + y) \] **Factorized Form:** \((b - a)(x + y)\) 3. **\( 3 a^{2} - a x + 6 a b - 2 b x \)** Factor by grouping: \[ 3a(a + 2b) - x(a + 2b) = (a + 2b)(3a - x) \] **Factorized Form:** \((3a - x)(a + 2b)\) 4. **\( x^{3} + 3x^{2} - x - 3 \)** Factor by grouping: \[ x^{2}(x + 3) - 1(x + 3) = (x + 3)(x^{2} - 1) = (x + 3)(x + 1)(x - 1) \] **Factorized Form:** \((x + 3)(x + 1)(x - 1)\) 5. **\( 2 a - 1 - 2 a b + b \)** Rearrange and factor: \[ (2a - 2ab) + (b - 1) = 2a(1 - b) + 1(b - 1) = (2a - 1)(1 - b) \] **Factorized Form:** \((2a - 1)(1 - b)\) 6. **\( 6 x^{2} - a y - 2 a x + 3 x y \)** Factor by grouping: \[ 2x(3x - a) + y(3x - a) = (3x - a)(2x + y) \] **Factorized Form:** \((3x - a)(2x + y)\) 7. **\( 4 a^{2} - 4 b^{2} - b^{3} + a^{2} b \)** Factor by grouping: \[ (4a^{2} + a^{2}b) - (4b^{2} + b^{3}) = a^{2}(4 + b) - b^{2}(4 + b) = (4 + b)(a^{2} - b^{2}) = (4 + b)(a - b)(a + b) \] **Factorized Form:** \((b + 4)(a - b)(a + b)\) 8. **\( 6 x^{2} - x^{5} - 6 + x^{3} \)** Rearrange and factor: \[ -x^{5} + x^{3} + 6x^{2} - 6 = -(x^5 - x^3 - 6x^2 + 6) \] Factor \(x^5 - x^3 - 6x^2 + 6\) by finding roots: \[ x^5 - x^3 - 6x^2 + 6 = (x - 1)(x + 1)(x^3 - 6) \] **Factorized Form:** \(- (x - 1)(x + 1)(x^3 - 6)\) 9. **\( x^{4} - x^{2} - x^{2} y^{2} + y^{2} \)** Factor by grouping: \[ x^{4} - x^{2} y^{2} - x^{2} + y^{2} = (x^{2} - 1)(x^{2} - y^{2}) = (x - 1)(x + 1)(x - y)(x + y) \] **Factorized Form:** \((x - 1)(x + 1)(x - y)(x + y)\) 10. **\( x^{3} + y^{3} + x + y \)** Factor by grouping: \[ x^{3} + y^{3} + x + y = (x + y)(x^{2} - xy + y^{2}) + (x + y) = (x + y)(x^{2} - xy + y^{2} + 1) \] **Factorized Form:** \((x + y)(x^{2} - xy + y^{2} + 1)\)