If (2x - 3) is a factor of 6x² + x + a, find the value of a. With this value of a, factorise the given expression.

By factor theorem (x - b) is a factor of f(x), if f(b) = 0.

f(x) = 6x² + x + a

Given, (2x - 3) or 2(x - $\Big(\dfrac{3}{2}\Big)$) is a factor of f(x) hence, f$\Big(\dfrac{3}{2}\Big)$ = 0.

$\therefore 6\Big(\dfrac{3}{2}\Big)^2 + \Big(\dfrac{3}{2}\Big) + a = 0 \\[1em] \Rightarrow 6\Big(\dfrac{9}{4}\Big) + \dfrac{3}{2} + a = 0 \\[1em] \Rightarrow \dfrac{27}{2} + \dfrac{3}{2} + a = 0 \\[1em] \Rightarrow \dfrac{30}{2} + a = 0 \\[1em] \Rightarrow 15 + a = 0 \\[1em] a = -15.$

Putting, a = -15 in f(x) we get,

f(x) = 6x² + x - 15

$\Rightarrow 6x^2 + 10x - 9x - 15 \\[0.5em] \Rightarrow 2x(3x + 5) - 3(3x + 5) \\[0.5em] (2x - 3)(3x + 5)$

Hence, the value of a is -15; 6x² + x - 15 = (2x - 3)(3x + 5).