Answer
The partial fraction is, $\frac{6}{\left( x-1 \right)}-\frac{5}{{{\left( x-1 \right)}^{2}}}$
Work Step by Step
$\frac{6x-11}{{{\left( x-1 \right)}^{2}}}=\frac{A}{\left( x-1 \right)}+\frac{B}{{{\left( x-1 \right)}^{2}}}$
Now, multiply both sides by ${{\left( x-1 \right)}^{2}}$:
${{\left( x-1 \right)}^{2}}\times \frac{6x-11}{{{\left( x-1 \right)}^{2}}}={{\left( x-1 \right)}^{2}}\times \frac{A}{\left( x-1 \right)}+{{\left( x-1 \right)}^{2}}\times \frac{B}{{{\left( x-1 \right)}^{2}}}$
$\begin{align}
& 6x-11=A\left( x-1 \right)+B \\
& =Ax-A+B
\end{align}$
Then, compare the coefficient of $x$ and constant term:
$A=6$ ….. (1)
$-A+B=-11$ …… (2)
And put the value of A in equation (2):
$\begin{align}
& -6+B=-11 \\
& B=-11+6 \\
& =-5
\end{align}$
Therefore,
$\frac{6x-11}{{{\left( x-1 \right)}^{2}}}=\frac{6}{\left( x-1 \right)}-\frac{5}{{{\left( x-1 \right)}^{2}}}$
Thus, the partial fraction of the provided expression is $\frac{6}{\left( x-1 \right)}-\frac{5}{{{\left( x-1 \right)}^{2}}}$.
