Matrices and determinants question. Establish that if A is the matrix 
\begin{bmatrix}
    b+c & a^2 & a \\
    c+a & b^2 & b \\
    a+b & c^2 & c \\
  \end{bmatrix}
then $|A| = -(a-b)(b-c)(c-a)(a+b+c)$.

using linearity with respect to the first column, those matrix have the same determinant:
\begin{bmatrix}
    b+c & a^2 & a \\
    c+a & b^2 & b \\
    a+b & c^2 & c \\
  \end{bmatrix}
\begin{bmatrix}
    b+c + a& a^2 & a \\
    c+a + b& b^2 & b \\
    a+b +c & c^2 & c \\
  \end{bmatrix} whose determinant is $a+b +c$ multiplied by
\begin{bmatrix}
    1 & a^2 & a \\
    1 & b^2 & b \\
    1 & c^2 & c \\
  \end{bmatrix} which is  $-1 \times$ the Vandermonde determinant.
