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. 

Hint. Is it possible to find $A,B,C$ such that for all $x\in\mathbb{R}$,
$$(x+3)^2=A(x+2)^2+B(x+1)^2+Cx^2?$$
P.S. The answer is yes: $A = 3$, $B = -3$, $C = 1$.
