Answer
$\frac{x^2-18x-30}{(5x+6)(x-2)}$.
Work Step by Step
The given expression is
$=\frac{3}{5x+6}-\frac{4}{x-2}+\frac{x^2-x}{5x^2-4x-12}$
First denominator $=(5x+6)$.
Second denominator $=x-2$.
Third denominator $=5x^2-4x-12$.
Rewrite the middle term $-4x$ as $-10x+6x$
$=5x^2-10x+6x-12$
Group terms.
$=(5x^2-10x)+(6x-12)$
Factor each term.
$=5x(x-2)+6(x-2)$
Factor out $(x-2)$.
$=(x-2)(5x+6)$
$=\frac{3}{5x+6}-\frac{4}{x-2}+\frac{x^2-x}{(x-2)(5x+6)}$
The LCM of all the denominators is $=(x-2)(5x+6)$.
$=\frac{3}{5x+6}\times \frac{x-2}{x-2}-\frac{4}{x-2}\times \frac{5x+6}{5x+6}+\frac{x^2-x}{(x-2)(5x+6)}$
Simplify.
$=\frac{3(x-2)}{(5x+6)(x-2)}-\frac{4(5x+6)}{(x-2)(5x+6)}+\frac{x^2-x}{(x-2)(5x+6)}$
$=\frac{3(x-2)-4(5x+6)+x^2-x}{(5x+6)(x-2)}$
$=\frac{3x-6-20x-24+x^2-x}{(5x+6)(x-2)}$
$=\frac{x^2-18x-30}{(5x+6)(x-2)}$.
