Intelligent choice of basis

Intelligent choice of basis

Let $V$ be the vector space of all polynomials of degrees $\le 3$. Let $U$
be the subspace of polynomials of the form $a_3 x^3 + a_2 x^2$. Say you
want to compute an orthogonal projection of some vector. I did a concrete
example by picking any basis for $V$ and then using Gram Schmidt procedure
to get an orthonormal basis. It was a long computation.
Is there an intelligent way to pick an orthonormal basis for $V$?
(intelligent=not blindly applying an algorithm invlolving so many
calculations like Gram Schmidt)