first, we have to see how this relates to the matrix A.

what does the matrix A do to a polynomial p(x) = ax^{2}+bx+c = [a,b,c] in the basis {x^{2},x,1}?

it takes [a,b,c] to [a-b+c,a+b+c,25a+5b+c].

note that p(-1) = a(-1)^{2}+ b(-1) + c = a-b+c

p(1) = a+b+c

p(5) = a(5_{2}) + b(5) + c = 25a+5b+c.

so A takes p(x) to q(x) = p(-1)x^{2}+ p(1)x + p(5).

suppose we are just given q(x), then to find p(x) we take A^{-1}(q(x)).

we are told by the problem that q(x) = -x^{2}+3x+31, which in our basis is [-1,3,31]. find A^{-1}of this vector.

(the fact that A is invertible means there is only ONE polynomial p(x) in P_{2}with this property).