If $f(x)$ and $g(x)$ are arbitrary polynomials of degree at most 2, then the mapping $$\langle f,g\rangle = f(-3)g(-3) + f(0)g(0) + f(2)g(2)$$ defines an inner product in $P_2$. Use this inner product to find $\langle f,g\rangle$, $\| f \|$, $\| g \|$, and the angle $\alpha_{f,g}$ between $f(x)$ and $g(x)$ for $$f(x) = 2 x^2 + 5 x + 9 \ \mbox{ and } \ g(x) = 3 x^2 - 5 x + 4.$$
$\langle f,g\rangle =$ ,
$\| f \| =$ ,
$\| g \| =$ ,
$\alpha_{f,g} =$ .