Prove Nice Determinant Equations I often come across these kinds of problems in A-level exam papers:

Prove that 
  $$
\begin{vmatrix}
    (a+b)^2 & 1       & 1      \\
    a^2     & (1+b)^2 & a^2    \\
    b^2     & b^2     & (1+a)^2 
\end{vmatrix} 
= 2ab(1+a+b)^3
$$

or 

Prove that 
  $$
\begin{vmatrix}
    1  & 1  & 1 \\
    x  & y  & z \\
    yz & xz & xy 
\end{vmatrix} 
= (x-y)(y-z)(z-x)
$$

Questions which involve nice determinant results, but which are quite a pain to prove by expanding. 
Are there any tricks that one can use to prove such results? I'm familiar with Vandermonde matrices for example, but I haven't come across anything that might help me with these, especially the first one.

By subtracting the first column from the second and the third we get that :
$$
\begin{split}
\begin{vmatrix}
    1  & 1  & 1 \\
    x  & y  & z \\
    yz & xz & xy 
\end{vmatrix}
&=\begin{vmatrix}
    1  & 0  & 0 \\
    x  & y-x  & z-x \\
    yz & z(x-y) & y(x-z) 
\end{vmatrix}\\
\\
&=(x-y)(z-x)\begin{vmatrix}
    1  & 0  & 0 \\
    x  & -1  & 1 \\
    yz & z & -y 
\end{vmatrix}\\
\\
&=(x-y)(z-x)\begin{vmatrix}
     -1  & 1 \\
     z & -y 
\end{vmatrix}\\
\\
& =(x-y)(y-z)(z-x)
\end{split}$$
