Factorise $27x^2 - 12$ completely. Hence, simplify $\dfrac{5}{27x^2 - 12} + \dfrac{2}{2 - 3x}$.

1. Extract the common factor from both terms.

$27x^2-12=3(9x^2-4)$

2. Factorise the remaining expression using difference of two squares.

$3(9x^2-4)=3\left((3x)^2-2^2\right)=3(3x+2)(3x-2)$

1. Substitute the factorised form from part (a) and make the second denominator compatible.

From part (a),
$27x^2-12=3(3x+2)(3x-2).$
Also,
$2-3x=-(3x-2).$
So
$\frac{5}{27x^2-12}+\frac{2}{2-3x}=\frac{5}{3(3x+2)(3x-2)}-\frac{2}{3x-2}.$

2. Express both fractions over a common denominator.

The common denominator is
$3(3x+2)(3x-2).$
Hence,

3. Combine and simplify the numerator.

$=\frac{5-18x-12}{3(3x+2)(3x-2)}$
$=\frac{-18x-7}{3(3x+2)(3x-2)}$
$=-\frac{18x+7}{3(3x+2)(3x-2)}.$