Find polynomial of smallest norm and of degree $\le 3$ for which $p(0)=2$
First use the fact that $p(0)=2$ , giving : $p(x)=2+bx+cx^2+dx^3$ .
Next calculate the integral:
$$
\int_{-1}^{+1} \left[ 2+bx+cx^2+dx^3 \right]^2\,dx =
8 + \frac{2}{7} d^2 + \frac{4}{5} b d + \frac{2}{5} c^2 + \frac{8}{3} c + \frac{2}{3} b^2
$$
Then make squares everywhere you can:
$$
= \frac{32}{9} + \frac{2}{5}\left(c+\frac{10}{3}\right)^2 + \frac{2}{7}\left(d+\frac{7}{5}b\right)^2 + \frac{8}{75}b^2
$$
Now it's easy to see that this expression is minimal for $b=d=0$ and $c=-10/3$ .
So here comes the polynomial that is asked for (with norm squared $=32/9$) :
$$
p(x)=2-\frac{10}{3}x^2
$$