a) \(x+x^2=0\Leftrightarrow x\left(1+x\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)
b) \(x+1-\left(x+1\right)^2=0\Leftrightarrow\left(x+1\right)\left(1-x-1\right)=0\)
\(\Leftrightarrow-x\left(x+1\right)\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)
c) \(15y\left(4y-9\right)-3\left(4y-9\right)=0\Leftrightarrow\left(15y-3\right)\left(4y-9\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{3}{15}=\dfrac{1}{5}\\x=\dfrac{9}{4}\end{matrix}\right.\)
d) \(8\left(25z+7\right)-27z\left(25z+7\right)=0\Leftrightarrow\left(8-27z\right)\left(25z+7\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}z=\dfrac{8}{27}\\z=\dfrac{-7}{25}\end{matrix}\right.\)