Answer
\[\frac{x+1}{(x - 3)(x + 2)}=\frac{4}{5(x-3)}+\frac{1}{5(x+2)}\]
Work Step by Step
In this case the general form of the partial fraction decomposition is
\[\frac{x+1}{(x - 3)(x + 2)}=\frac{A}{x-3}+\frac{B}{x+2}\;\;\;...(*)\]
\[\Rightarrow x+1=A(x+2)+B(x-3)\]
\[\Rightarrow x+1=(A+B)x+(2A-3B)\]
Comparing like coefficients
$A+B=1$ _____(1)
$2A-3B=1$ ____(2)
Multiply equation (1) by 3 then add to equation (2)
$\Rightarrow 5A=4\Rightarrow A=\frac{4}{5}$
From (1)
$\Rightarrow B=\frac{1}{5}$
From (*)
\[\frac{x+1}{(x - 3)(x + 2)}=\frac{4}{5(x-3)}+\frac{1}{5(x+2)}\]
Hence,
$\frac{x+1}{(x - 3)(x + 2)}=\frac{4}{5(x-3)}+\frac{1}{5(x+2)}$.
