Answer
$\displaystyle \frac{7}{x-9}-\frac{4}{x+2}$
Work Step by Step
Both factors of the denominator are distinct linear factors...
see p.541. Decomposition with Distinct Linear Factors In the Denominator:
Include one partial fraction with a $constant$ numerator for each distinct linear factor in the denominator.
Setup:
$\displaystyle \frac{3x+50}{(x-9)(x+2)}=\frac{A}{x-9}+\frac{B}{x+2}\qquad /\times(x-9)(x+2)\quad$(LCD)
$3x+50=A(x+2)+B(x-9)$
$ 3x+50=Ax+2A+Bx-9B\qquad$ /group like terms...
$3x+50=x(A+B)+(2A-9B)$
... we now equate the coefficients of like powers...
$\left\{\begin{array}{l}
3=A+B\\
50=2A-9B
\end{array}\right.\qquad $solve the system
Multiply equation 1. with $-2$ and add the equations:
$-6+50=-2A+2A-2B-9B$
$44=-11B$
$B=-4$
back substitute: $3=A+B$
$A=3-B$
$A=3-(-4)$
$A=7$
Rewrite the setup decomposition with $A=7, B=-4$
$\displaystyle \frac{3x+50}{(x-9)(x+2)}=\frac{7}{x-9}-\frac{4}{x+2}$
