<s>[INST] Calculate complex determinant $$\left| {\begin{array}{*{20}{c}}{{a^2}}&{{{(a + 1)}^2}}&{{{(a + 2)}^2}}&{{{(a + 3)}^2}}\\{{b^2}}&{{{(b + 1)}^2}}&{{{(b + 2)}^2}}&{{{(b + 3)}^2}}\\{{c^2}}&{{{(c + 1)}^2}}&{{{(c + 2)}^2}}&{{{(c + 3)}^2}}\\{{d^2}}&{{{(d + 1)}^2}}&{{{(d + 2)}^2}}&{{{(d + 3)}^2}}\end{array}} \right|
$$
It's very stupid desicion for me to expand it by row or column. 
Any suggestions?
 [/INST] By the fourth column (resp. the third column, the second column) subtract the third column (resp. the second column, the first column), you get 
$\left|\begin{array}{cccc} 
a^2 & 2a+1 & 2a+3 & 2a+5 \\
b^2 & 2b+1 & 2b+3 & 2b+5 \\
c^2 & 2c+1 & 2c+3 & 2c+5 \\
d^2 & 2d+1 & 2d+3 & 2d+5 \\
\end{array} \right|=0$
</s>