Use the inner product...

Use the inner product
< f,g > = f(−2)g(−2) + f(0)g(0) + f(2)g(2)
in P2 to find the orthogonal projection of f(x) = 4x^2 + 4x + 1 onto the line L spanned by g(x) = 2x^2 − 5x + 8.
proj_L(f) = ?

I found the inner product to be 264 but I don't know what to do next.

On a one-dimensional subspace of an inner product space, the orthogonal projection of f onto the space L spanned by g is given by:

proj_L(f) = ( / )g

= f(-2)g(-2) + f(0)g(0) + f(2)g(2)
= 9 * 26 + 1 * 8 + 25 * 6
= 392

= g(-2)^2 + g(0)^2 + g(2)^2
= 26^2 + 8^2 + 6^2
= 776

So, the orthogonal projection is:

(392/776)g(x)
= (49/97)(2x^2 - 5x + 8)