Answer
$$\frac{3}{{x - 2}} + \frac{2}{{x - 1}}$$
Work Step by Step
$$\eqalign{
& \frac{{5x - 7}}{{{x^2} - 3x + 2}} \cr
& {\text{Factoring the denominator}} \cr
& \frac{{5x - 7}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} \cr
& {\text{The partial fraction decomposition has the form}} \cr
& \frac{{5x - 7}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} = \frac{A}{{x - 2}} + \frac{B}{{x - 1}}\,\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{Find the values of the coefficients }}A,B{\text{; we clear fractions to obtain}} \cr
& 5x - 7 = A\left( {x - 1} \right) + B\left( {x - 2} \right) \cr
& \,\,\,\,\,{\text{set }}x = 2 \cr
& 5\left( 2 \right) - 7 = A\left( {2 - 1} \right) + B\left( {2 - 2} \right) \cr
& 3 = A \cr
& \,\,\,\,\,{\text{set }}x = 1 \cr
& 5\left( 1 \right) - 7 = A\left( {1 - 1} \right) + B\left( {1 - 2} \right) \cr
& - 2 = B\left( { - 1} \right) \cr
& B = 2 \cr
& {\text{Then substituting }}A{\text{ and }}B{\text{ into the equation }}\left( {\bf{1}} \right) \cr
& \frac{{5x - 7}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} = \frac{3}{{x - 2}} + \frac{2}{{x - 1}} \cr} $$
