Write as a single fraction in its simplest form $\displaystyle \frac{4 - x}{x + 2} \; - \; \frac{2}{3 - x}$. Answer ...

1. Rewrite the subtraction using a common denominator.

$\frac{4-x}{x+2}-\frac{2}{3-x}$
Since $3-x=-(x-3)$,
$-\frac{2}{3-x}=\frac{2}{x-3}.$
So
$\frac{4-x}{x+2}-\frac{2}{3-x}=\frac{4-x}{x+2}+\frac{2}{x-3}.$

2. Combine into one fraction and simplify the numerator.

$=\frac{4x-12-x^2+3x+2x+4}{(x+2)(x-3)}$
$=\frac{-x^2+9x-8}{(x+2)(x-3)}.$
Alternatively, using denominator $(x+2)(3-x)$ directly gives