Answer
$ \frac{4(x-2)}{2x+5}$
Work Step by Step
The given expression is
$\Rightarrow \frac{\frac{4}{x+3}}{\frac{2}{x-2}-\frac{1}{x^2+x-6}}$
Factor $x^2+x-6$
Rewrite the middle term $x$ as $3x-2x$.
$\Rightarrow x^2+3x-2x-6$
Group the terms.
$\Rightarrow (x^2+3x)+(-2x-6)$
Factor each group.
$\Rightarrow x(x+3)-2(x+3)$
Factor out $(x+3)$.
$\Rightarrow (x+3)(x-2)$.
Substitute back the factor into the given expression.
$\Rightarrow \frac{\frac{4}{x+3}}{\frac{2}{x-2}-\frac{1}{(x+3)(x-2)}}$
The LCD is $(x+3)(x-2)$.
Multiply the numerator and the denominator by $(x+3)(x-2)$.
$\Rightarrow \frac{(x+3)(x-2)\left (\frac{4}{x+3}\right )}{(x+3)(x-2)\left (\frac{2}{x-2}-\frac{1}{(x+3)(x-2)}\right)}$
Use the distributive property.
$\Rightarrow \frac{(x+3)(x-2)\left (\frac{4}{x+3}\right )}{(x+3)(x-2)\left (\frac{2}{x-2}\right)-(x+3)(x-2)\left (\frac{1}{(x+3)(x-2)}\right)}$
Cancel common terms.
$\Rightarrow \frac{4(x-2)}{2(x+3)-1}$
Use the distributive property.
$\Rightarrow \frac{4(x-2)}{2x+6-1}$
Simplify.
$\Rightarrow \frac{4(x-2)}{2x+5}$
