Answer
\[\frac{x-2}{(x - 1)(x + 4)}=\frac{-1}{5(x-1)}+\frac{6}{5(x+4)}\]
Work Step by Step
The general form of the decomposition is
\[\frac{x-2}{(x - 1)(x + 4)}=\frac{A}{x-1}+\frac{B}{x+4}\;\;\;...(*)\]
\[\Rightarrow (x-2)=A(x+4)+B(x-1)\]
\[\Rightarrow x-2=(A+B)x+(4A-B)\]
Comparing like coefficients
$A+B=1\;\;\;...(1)$
$4A-B=-2\;\;\;...(2)$
Adding equation (1) and (2)
$5A=-1\Rightarrow A=\frac{-1}{5}$
From (1)
$\Rightarrow B=\frac{6}{5}$
From (*)
\[\frac{x-2}{(x - 1)(x + 4)}=\frac{-1}{5(x-1)}+\frac{6}{5(x+4)}\]
Hence,
$\frac{x-2}{(x - 1)(x + 4)}=\frac{-1}{5(x-1)}+\frac{6}{5(x+4)}$.
