Answer
Solution set = $\{2\}$.
Work Step by Step
The RHS denominator $x^{2}+7x+12$ is factored by searching
for two factors of $c=12$ whose sum is $b=7$ ... we find +4 and +3
$x^{2}+7x+12=(x+3)(x+4)$
First, we exclude those values of x that yield a zero in any of the denominators.
$x\not\in\{-4,-3 \}\qquad (*)$
Multiply the equation with the LCD=$(x+3)(x+4)$
$ 5(x+3)+3(x+4)=12x+19\qquad$ ... simplify (distribute)
$5x+15+3x+12=12x+19$
$ 8x+27=12x+19\qquad$ ... add $-8x-19$ to both sides
$8=4x$
$ x=2 \qquad$ ...Checking (*), this is a valid solution
Solution set = $\{2\}$.
