(a) Factorise: \( \begin{array}{llll}\text { (1) } 6 x^{3}+12 x & \text { (2) } 6 x^{3}+4 x^{2} & \text { (3) } 5 x^{3}+5 x \\ \text { (4) } 12 x^{3}-18 x^{2} & \text { (5) } & 3 x^{2}-9 y+12 x y & \text { (6) } \\ 8 a^{2} b^{2}-64 a b \\ \text { (7) } 16 m^{4} n^{8}-8 m^{3} n^{7}+36 m^{2} n^{3} & & & \end{array} \)

Let's tackle the factorization for each expression in your list: **(1) \( 6x^{3} + 12x \)** Factor out the common term \( 6x \): \[ 6x(x^2 + 2) \] **(2) \( 6x^{3} + 4x^{2} \)** Factor out the common term \( 2x^{2} \): \[ 2x^{2}(3x + 2) \] **(3) \( 5x^{3} + 5x \)** Factor out the common term \( 5x \): \[ 5x(x^{2} + 1) \] **(4) \( 12x^{3} - 18x^{2} \)** Factor out the common term \( 6x^{2} \): \[ 6x^{2}(2x - 3) \] **(5) \( 3x^{2} - 9y + 12xy \)** Rearranging gives \( 3x^{2} + 12xy - 9y \). Factor out \( 3 \): \[ 3(x^{2} + 4xy - 3y) \] **(6) \( 8a^{2}b^{2} - 64ab \)** Factor out the common term \( 8ab \): \[ 8ab(ab - 8) \] **(7) \( 16m^{4}n^{8} - 8m^{3}n^{7} + 36m^{2}n^{3} \)** This expression can be quite intricate, but we can first factor out the common term \( 4m^{2}n^{3} \): \[ 4m^{2}n^{3}(4m^{2}n^{5} - 2mn^{4} + 9) \] Now, you've got the factorized forms neatly lined up! Happy factorizations!