Need help with a determinant problem

Question

I'm learning determinants and just came across a problem. I've been trying really hard to solve it but no success so far. I just know that the answer is (3) 1 but don't know how to solve it? Please explain in details.

Thank you.

Answer 1

$$\scriptsize\begin{vmatrix} 3&1+a+b&1+a^2+b^2\\ 1+a+b&1+a^2+b^2&1+a^3+b^3\\ 1+a^2+b^2&1+a^3+b^3&1+a^4+b^4 \end{vmatrix}=\begin{vmatrix} 1&1&1\\ 1&a&b\\ 1&a^2&b^2 \end{vmatrix}\begin{vmatrix} 1&1&1\\ 1&a&a^2\\ 1&b&b^2 \end{vmatrix}=\begin{vmatrix} 1&1&1\\ 1&a&b\\ 1&a^2&b^2 \end{vmatrix}^2=[(a-1)(b-1)(a-b)]^2$$