\begin{aligned}
\frac{5 y-12}{(y-2)(y-3)} &=\frac{A}{y-2}+\frac{B}{y-3} \\
\frac{5 y-12}{(1-2)(y-3)} &=\frac{A(y-3)+B(y-2)}{(y-2)(y-3)} \\
5 y-12 &=A(y-3)+B(y-2)
\end{aligned}
\begin{aligned}
\text { Put } y &=3 \\
5(3)-12 &=A(3-3)+B(3-2) \\
3 &=B \\
\text { Again. put } y &=2 \\
5(2)-12 &=A(2-3)+B(2-2) \\
-2 &=-A, A=2 \\
\frac{5 y-12}{(y-2)(y-3)} &=\frac{2}{y-2}+\frac{3}{y-3}
\end{aligned}