Answer
$\dfrac{x^3+x^2+2}{(x^2+2)^2}=\dfrac{x+1}{x^2+2}-\dfrac{2x}{(x^2+2)^2}$
Work Step by Step
We are given the fraction:
$\dfrac{x^3+x^2+2}{(x^2+2)^2}$
As the denominator is factored, we can write the partial fraction decomposition:
$\dfrac{x^3+x^2+2}{(x^2+2)^2}=\dfrac{Ax+B}{x^2+2}+\dfrac{Cx+D}{(x^2+2)^2}$
Multiply all terms by the least common denominator $(x^2+2)^2$:
$(x^2+2)^2\cdot\dfrac{x^3+x^2+2}{(x^2+2)^2}=(x^2+2)^2\cdot\dfrac{Ax+B}{x^2+2}+(x^2+2)^2\cdot\dfrac{Cx+D}{(x^2+2)^2}$
$x^3+x^2+2=(Ax+B)(x^2+2)+(Cx+D)$
$x^3+x^2+2=Ax^3+2Ax+Bx^2+2B+Cx+D$
$x^3+x^2+2=Ax^3+Bx^2+(2A+C)x+(2B+D)$
Identify the coefficients and build the system of equations:
$\begin{cases}
A=1\\
B=1\\
2A+C=0\\
2B+D=2
\end{cases}$
Solve the system:
$A=1$
$B=1$
$2A+C=0$
$2(1)+C=0$
$C=-2$
$2B+D=2$
$2(1)+D=2$
$2+D=2$
$D=0$
The partial fraction decomposition is:
$\dfrac{x^3+x^2+2}{(x^2+2)^2}=\dfrac{x+1}{x^2+2}-\dfrac{2x}{(x^2+2)^2}$
