Answer
$$\frac{3}{{x - 2}} + \frac{2}{{x - 1}}$$
Work Step by Step
$$\eqalign{
& \frac{{5x - 7}}{{{x^2} - 3x + 2}} \cr
& {\text{factoring}} \cr
& = \frac{{5x - 7}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} \cr
& {\text{partial fraction decomposition}} \cr
& \frac{{5x - 7}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} = \frac{A}{{x - 2}} + \frac{B}{{x - 1}} \cr
& 5x - 7 = A\left( {x - 1} \right) + B\left( {x - 2} \right) \cr
& {\text{letting }}x = 2 \cr
& 5\left( 2 \right) - 7 = A\left( {2 - 1} \right) + B\left( {2 - 2} \right) \cr
& 3 = A \cr
& {\text{letting }}x = 1 \cr
& 5\left( 1 \right) - 7 = A\left( {1 - 1} \right) + B\left( {1 - 2} \right) \cr
& - 2 = B\left( { - 1} \right) \cr
& B = 2 \cr
& {\text{substituting the values}} \cr
& \frac{A}{{x - 2}} + \frac{B}{{x - 1}} = \frac{3}{{x - 2}} + \frac{2}{{x - 1}} \cr} $$
