Answer
$\{-\frac{9}{2}\}$.
Work Step by Step
Determine the Least Common Denominator ($LCD$) so that we clear fractions.
Factor $x^2+3x+2$.
Rewrite the middle term $3x$ as $2x+1x$.
$\Rightarrow x^2+2x+1x+2$
Group the terms.
$\Rightarrow (x^2+2x)+(1x+2)$
Factor each group.
$\Rightarrow x(x+2)+1(x+2)$
Factor out $(x+2)$.
$\Rightarrow (x+2)(x+1)$.
Back substitute the factor into the given equation.
$\Rightarrow \frac{x+5}{x+1}-\frac{x}{x+2}=\frac{4x+1}{(x+2)(x+1)}$
Multiply the equation by $LCD=(x+2)(x+1)$.
$\Rightarrow (x+2)(x+1)\left ( \frac{x+5}{x+1}-\frac{x}{x+2}\right )=(x+2)(x+1)\left (\frac{4x+1}{(x+2)(x+1)}\right )$
Use the distributive property and cancel common factors.
$\Rightarrow (x+2)(x+5)-(x)(x+1)=4x+1$
Use the FOIL method and the distributive property.
$\Rightarrow x^2+5x+2x+10-x^2-x=4x+1$
Add like terms.
$\Rightarrow 6x+10=4x+1$
Add $-4x-10$ to both sides.
$\Rightarrow 6x+10-4x-10=4x+1-4x-10$
Simplify.
$\Rightarrow 2x=-9$
Divide both sides by $2$.
$\Rightarrow \frac{2x}{2}=\frac{-9}{2}$
Simplify.
$\Rightarrow x=-\frac{9}{2}$
The solution set is $\{-\frac{9}{2}\}$.
