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.** \( t x - t y + r x - r y \) **Factorization:** \[ t x - t y + r x - r y = t(x - y) + r(x - y) = (t + r)(x - y) \] --- **2.** \( b y - a x + b x - a y \) **Factorization:** \[ b y - a x + b x - a y = b y - a y + b x - a x = y(b - a) + x(b - a) = (b - a)(x + y) \] Alternatively, factoring out \(-1\) from \((b - a)\): \[ = -(a - b)(x + y) \] --- **3.** \( 3 a^{2} - a x + 6 a b - 2 b x \) **Factorization:** \[ 3a^2 - ax + 6ab - 2bx = 3a^2 + 6ab - ax - 2bx = 3a(a + 2b) - x(a + 2b) = (3a - x)(a + 2b) \] --- **4.** \( x^{3} + 3 x^{2} - x - 3 \) **Factorization:** \[ x^3 + 3x^2 - x - 3 = x^2(x + 3) -1(x + 3) = (x^2 -1)(x + 3) = (x -1)(x +1)(x + 3) \] --- **5.** \( 2 a - 1 - 2 a b + b \) **Factorization:** \[ 2a - 1 - 2ab + b = 2a(1 - b) + 1(b -1) = (2a +1)(1 - b) \] Alternatively: \[ = (2a + 1)(- (b -1)) = -(2a +1)(b -1) \] --- **6.** \( 6 x^{2} - a y - 2 a x + 3 x y \) **Factorization:** Rearrange terms: \[ 6x^2 - 2ax + 3xy - ay \] Factor by grouping: \[ (6x^2 -2ax) + (3xy - ay) = 2x(3x - a) + y(3x - a) = (2x + y)(3x - a) \] --- **7.** \( 4 a^{2} - 4 b^{2} - b^{3} + a^{2} b \) **Factorization:** Rearrange and group: \[ 4a^2 + a^2b - 4b^2 - b^3 = a^2(4 + b) - b^2(4 + b) = (a^2 - b^2)(4 + b) = (a - b)(a + b)(4 + b) \] --- **8.** \( 6 x^{2} - x^{5} -6 +x^{3} \) **Factorization:** Rearrange: \[ -x^5 + x^3 + 6x^2 -6 = -x^5 + x^3 +6x^2 -6 \] Factor by grouping: \[ (-x^5 + x^3) + (6x^2 -6) = -x^3(x^2 -1) +6(x^2 -1) = ( -x^3 +6)(x^2 -1) \] Factor further: \[ (x^2 -1) = (x -1)(x +1) \] Thus: \[ (-x^3 +6)(x -1)(x +1) \quad \text{or} \quad -(x^3 -6)(x -1)(x +1) \] --- **9.** \( x^{4} -x^{2} -x^{2} y^{2} + y^{2} \) **Factorization:** Rearrange: \[ x^4 - x^2 y^2 -x^2 + y^2 = x^2(x^2 - y^2 -1) + y^2 \] Notice that \(x^2 - y^2 = (x - y)(x + y)\), but let's try factoring by grouping: \[ x^4 - x^2 y^2 - x^2 + y^2 = (x^4 - x^2 y^2) - (x^2 - y^2) =x^2(x^2 - y^2) -1(x^2 - y^2) = (x^2 -1)(x^2 - y^2) = (x -1)(x +1)(x - y)(x + y) \] --- **10.** \( x^{3} + y^{3} + x + y \) **Factorization:** Recall that \(x^3 + y^3 = (x + y)(x^2 -xy + y^2)\). Adding \(x + y\): \[ x^3 + y^3 + x + y = (x + y)(x^2 -xy + y^2) + (x + y) = (x + y)(x^2 - xy + y^2 +1) \] Alternatively, factor out \((x+y)\): \[ (x + y)(x^2 - xy + y^2 +1) \] Thus, the complete factorization is: \[ (x + y)(x^2 - xy + y^2 +1) \] --- **Summary of All Factorizations:** 1. \((t + r)(x - y)\) 2. \((b - a)(x + y)\) 3. \((3a - x)(a + 2b)\) 4. \((x -1)(x +1)(x + 3)\) 5. \((2a +1)(1 - b)\) 6. \((2x + y)(3x - a)\) 7. \((a - b)(a + b)(4 + b)\) 8. \(-(x^3 -6)(x -1)(x +1)\) 9. \((x -1)(x +1)(x - y)(x + y)\) 10. \((x + y)(x^2 - xy + y^2 +1)\)