Answer
The partial fraction is $\frac{a}{x-c}+\frac{ac+b}{{{\left( x-c \right)}^{2}}}$.
Work Step by Step
We know that the partial fraction can be written as given below,
$\begin{align}
& \frac{ax+b}{{{\left( x-c \right)}^{2}}}=\frac{A}{x-c}+\frac{B}{{{\left( x-c \right)}^{2}}} \\
& =\frac{A\left( x-c \right)+B}{{{\left( x-c \right)}^{2}}}
\end{align}$
And,
$ax+b=A\left( x-c \right)+B$ …… (1)
Substitute the value of $x=c$ in equation (1),
$ac+b=B$
Again, put the value of $x=0$ in equation (1),
$\begin{align}
& b=-Ac+B \\
& b=-Ac+ac+b \\
& Ac=ac \\
& A=a
\end{align}$
Therefore,
$\frac{ax+b}{{{\left( x-c \right)}^{2}}}=\frac{a}{x-c}+\frac{ac+b}{{{\left( x-c \right)}^{2}}}$
Thus, the partial fraction decomposition form is $\frac{ax+b}{{{\left( x-c \right)}^{2}}}=\frac{a}{x-c}+\frac{ac+b}{{{\left( x-c \right)}^{2}}}$.
