\(\frac{4x^2-16}{x^2+2x}=\frac{A}{x}\)
\(\Rightarrow A.\left(x^2+2x\right)=\left(4x^2-16\right).x\)
\(\Rightarrow A=\frac{\left[\left(2x\right)^2-4^2\right].x}{x^2+2x}\)
\(A=\frac{\left(2x-4\right)\left(2x+4\right).x}{x\left(x+2\right)}\)
\(A=\frac{2.2.\left(x-2\right)\left(x+2\right).x}{x\left(x+2\right)}\)
\(A=4\left(x-2\right)\)\(\left(x\ne0;x+2\ne0\right)\)
\(A=4x-8\)