#### Answer

The simplified partial fraction expansion is $\frac{5x+7}{\left( x-1 \right)\left( x+3 \right)}=\frac{A}{\left( x-1 \right)}+\frac{B}{\left( x+3 \right)}$

#### Work Step by Step

The provided rational expression is as follows:
$\frac{5x+7}{\left( x-1 \right)\left( x+3 \right)}$
Now, solving the expression as given below:
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.
Then, decompose the fractional part as follows:
$\frac{5x+7}{\left( x-1 \right)\left( x+3 \right)}=\frac{A}{\left( x-1 \right)}+\frac{B}{\left( x+3 \right)}$
Thus, $\frac{A}{\left( x-1 \right)}+\frac{B}{\left( x+3 \right)}$ is a partial fraction expansion of the rational expression $\frac{5x+7}{\left( x-1 \right)\left( x+3 \right)}$ with constants $A$ and $B$.