<s>[INST] What will be the value of the following determinant without expanding it? $$\begin{vmatrix}a^2 & (a+1)^2 & (a+2)^2 & (a+3)^2 \\ b^2 & (b+1)^2 & (b+2)^2 & (b+3)^2 \\ c^2 & (c+1)^2 & (c+2)^2 & (c+3)^2 \\ d^2 & (d+1)^2 & (d+2)^2 & (d+3)^2\end{vmatrix} $$
I tried many column operations, mainly subtractions without any success. 
 [/INST] The determinant is a polynom of the second degree with respect to $a$, on the other hand, it has three different (in general) roots $b$, $c$, and $d$ (as, e.g., if $a=b$, the determinant has two identical rows), therefore, the determinant vanishes.
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