$a,\left(x-4\right)^2-25=0$

$\Rightarrow\left(x-4-5\right)\left(x-4+5\right)=0$

$\Rightarrow\left(x-9\right)\left(x+1\right)=0$

$\Rightarrow\left[{}\begin{matrix}x-9=0\\x+1=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=9\\x=-1\end{matrix}\right.$

$b,\left(x-3\right)^2-\left(x+1\right)^2=0$

$\Rightarrow\left(x-3-x-1\right)\left(x-3+x+1\right)=0$

$\Rightarrow-4\left(2x-2\right)=0$

$\Rightarrow2\left(x-1\right)=0$

$\Rightarrow x-1=0$

$\Rightarrow x=1$

$c,\left(x^2-4\right)\left(2x+3\right)=\left(x^2-4\right)\left(x-1\right)$

$\Rightarrow2x+3=x-1$

$\Rightarrow2x-x=-1-3$

$\Rightarrow x=-4$

$d,\left(3x-7\right)^2-4\left(x+1\right)^2=0$

$\Rightarrow\left(3x-7\right)-\left[2\left(x+1\right)\right]^2=0$

$\Rightarrow\left(3x-7\right)^2-\left(2x+2\right)^2=0$

$\Rightarrow\left(3x-7-2x-2\right)\left(3x-7+2x+2\right)=0$

$\Rightarrow\left(x-9\right)\left(5x-5\right)=0$

$\Rightarrow5\left(x-9\right)\left(x-1\right)=0$

$\Rightarrow\left[{}\begin{matrix}x-9=0\\x-1=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=9\\x=1\end{matrix}\right.$