a.
\(\dfrac{A}{x^2-1}=\dfrac{1}{x-1}\Rightarrow A\left(x-1\right)=x^2-1\)
\(\Rightarrow A\left(x-1\right)=\left(x-1\right)\left(x+1\right)\)
\(\Rightarrow A=x+1\)
b.
\(\dfrac{x^2+2x}{A}=\dfrac{x}{3}\Rightarrow A.x=3\left(x^2+2x\right)\)
\(\Rightarrow A.x=3x.\left(x+2\right)\)
\(\Rightarrow A=3\left(x+2\right)\)
c.
\(\dfrac{x-3}{x^2-9}=\dfrac{A}{x+3}\Rightarrow A\left(x^2-9\right)=\left(x-3\right)\left(x+3\right)\)
\(\Rightarrow A\left(x-3\right)\left(x+3\right)=\left(x-3\right)\left(x+3\right)\)
\(\Rightarrow A=1\)
d.
\(\dfrac{x^2+3x-4}{A}=x+4\Rightarrow A\left(x+4\right)=x^2+3x-4\)
\(\Rightarrow A\left(x+4\right)=\left(x-1\right)\left(x+4\right)\)
\(\Rightarrow A=x-1\)
