Answer
$\frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}=\frac{2x-5}{x^2+x+2}+\frac{5}{x^2+1}$
Work Step by Step
$\frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}=\frac{Ax+B}{x^2+x+2}+\frac{Cx+D}{x^2+1}$,
$Ax+B(x^2+1)+Cx+D(x^2+x+2)$,
$Ax^3+Ax+Bx^2+B+Cx^3+Cx^2+2Cx+Dx^2+Dx+2D$,
$(A+C)x^3+(B+C+D)x^2+(A+2C+D)x+(B+2D)=2x^3+7x+5$,
$\begin{cases}
A+C=2\\
B+C+D=0\\
A+2C+D=7\\
B+2D=5
\end{cases}$
Multiplying Equation 1 by -1 and adding it to Equation 3.
$\begin{cases}
-A-C=-2\\
A+2C+D=7\\
-- -- -- -- --\\
C+D=5
\end{cases}$
Multiplying Equation 2 by -1 and Equation 4.
$\begin{cases}
-B-C-D=0\\
B+2D=5\\
-- -- -- --\\
-C+D=5
\end{cases}$
Adding the addition results.
$\begin{cases}
C+D=5\\
-C+D=5\\
-- -- -- --\\
2D=10
\end{cases}$
thus, $D=5$, substituting back into the Equation, $C+5=5, C=0$.
substituting back into the Equation, $A+0=2, A=2$.
substituting back into the Equation $B+10=5, B=-5$.
Therefore,
$\frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}=\frac{2x-5}{x^2+x+2}+\frac{5}{x^2+1}$
