Use the inner product

in to find the orthogonal

projection of onto the line spanned by .

.

Which is wrong.

The projection of f onto L is given by f(x)-{g(x) dot f(x)/||g(x)||^2}*g(x)

The inner product of g(x) and f(x) is

(3 (-3)^2+6 (-3)+3) (2 (-3)^2-6 (-3)-9)+(3(0)^2 +6(0) +3)*(2(0)^2 -6(0) -9)+(3(3)^2 +6(3) +3)*(2(3)^2 -6(3) -9)= -135

Inner product of g(x) with itself is then:

(2(-3)^2 -6(-3) -9)*(2(-3)^2 -6(-3) -9)+(2(0)^2 -6(0) -9)*(2(0)^2 -6(0) -9)+(2(3)^2 -6(3) -9)*(2(3)^2 -6(3) -9) = 891

-135/891 simplifies to -5/33

Multiplying g(x) with -5/33 gives -(10 x^2)/33+(10 x)/11+15/11

Subtracting this value from f(x) results with

(109 x^2)/33+(56 x)/11+18/11, which is my final answer.

I greatly appreciate your help!