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. 

I wanted to present a different view of the problem that takes advantage of the sameness of the rows in a less computational way.
We can view the entries of each row as the evaluations of 4 polynomials of degree 2: $x^2$, $(x+1)^2$, $(x+2)^2$ and $(x+3)^2$ at a point.
Since the set of polynomials of degree at most 2, $\mathbb{P}_2$, forms a 3 dimensional vector space over $\mathbb{R}$, any collection of 4 polynomials will be linearly dependent.  Hence there is some dependence relation among the columns.  This dependence relation, $\alpha_0x^2+\alpha_1(x+1)^2+\alpha_2(x+2)^2+\alpha_3(x+3)^2=0$ means some combination of elementary column operations will create an all 0's column.  Hence the matrix has linearly dependent columns.
