#### Answer

The partial fraction expansion is $\frac{3x+16}{\left( x+1 \right){{\left( x-2 \right)}^{2}}}=\frac{A}{\left( x+1 \right)}+\frac{B}{\left( x-2 \right)}+\frac{C}{{{\left( x-2 \right)}^{2}}}$.

#### Work Step by Step

The provided partial expression is as given below:
$\frac{3x+16}{\left( x+1 \right){{\left( x-2 \right)}^{2}}}$
Now, solve the expression as follows:
We set up the partial fraction expansion with unknown constants coefficients and then write a constant coefficients over each of the two distinct algebraic linear factors in the denominator of the expression.
$\frac{3x+16}{\left( x+1 \right){{\left( x-2 \right)}^{2}}}=\frac{A}{\left( x+1 \right)}+\frac{B}{\left( x-2 \right)}+\frac{C}{{{\left( x-2 \right)}^{2}}}$
Thus, $\frac{A}{\left( x+1 \right)}+\frac{B}{\left( x-2 \right)}+\frac{C}{{{\left( x-2 \right)}^{2}}}$ is a partial fraction expansion of the rational expression $\frac{3x+16}{\left( x+1 \right){{\left( x-2 \right)}^{2}}}$ with constants $A$, $B$ and $C$.