$a)y=\frac{2x-3}{x+1}=\frac{2x+2-5}{x+1}=\frac{2\left(x+1\right)}{x+1}-\frac{5}{x+1}=\frac{-5}{x+1}+2$  

$\{\begin{array}{l}a=-5\\ p=-1\\ q=2\end{array}$  

$b)y=\frac{-3x-6}{x-2}=\frac{-3x+6-12}{x-2}=\frac{-3\left(x-2\right)-12}{x-2}=\frac{-12}{x-2}-3$  

$\{\begin{array}{l}a=-12\\ p=2\\ q=-3\end{array}$   

 

$c)y=\frac{4x+4}{8x+1}=\frac{4x+\frac{1}{2}+\frac{7}{2}}{8x+1}=\frac{\frac{1}{2}\left(8x+1\right)+\frac{7}{2}}{8x+1}=\frac{\frac{7}{2}}{8\left(x+\frac{1}{8}\right)}+\frac{1}{2}=\frac{\frac{7}{16}}{x+\frac{1}{8}}+\frac{1}{2}$  

$\{\begin{array}{l}a=\frac{7}{16}\\ p=-\frac{1}{8}\\ q=\frac{1}{2}\end{array}$  

 

$d)y=\frac{3x}{6x-3}=\frac{3x-\frac{3}{2}+\frac{3}{2}}{6x-3}=\frac{\frac{1}{2}\left(6x-3\right)+\frac{3}{2}}{6x-3}=\frac{\frac{3}{2}}{6\left(x-\frac{1}{2}\right)}+\frac{1}{2}=\frac{\frac{1}{4}}{x-\frac{1}{2}}+\frac{1}{2}$  

$\{\begin{array}{l}a=\frac{1}{4}\\ p=\frac{1}{2}\\ q=\frac{1}{2}\end{array}$  

 

$e)y=\frac{x-5}{4x}=\frac{x}{4x}-\frac{5}{4x}=\frac{-\frac{5}{4}}{x}+\frac{1}{4}$  

$\{\begin{array}{l}a=-\frac{5}{4}\\ p=0\\ q=\frac{1}{4}\end{array}$  

 

$f)y=\frac{2x-9}{5x-1}=\frac{2x-\frac{2}{5}-\frac{43}{5}}{5x-1}=\frac{\frac{2}{5}\left(5x-1\right)-\frac{43}{5}}{5x-1}=\frac{-\frac{43}{5}}{5\left(x-\frac{1}{5}\right)}+\frac{2}{5}=\frac{-\frac{43}{25}}{x-\frac{1}{5}}+\frac{2}{5}$  

$\{\begin{array}{l}a=-\frac{43}{25}\\ p=\frac{1}{5}\\ q=\frac{2}{5}\end{array}$