A question about subspace and finding basis

Question

guys help me I couldn't solve this question I've been working on subspaces for sometime but still cant do this kind of questions.

Answer 1

Hint

Prove that that the zero function belongs to $\mathcal U$ and if $f,g\in\mathcal U$ and $\lambda\in\Bbb R$ then $\lambda f+g\in \mathcal U$.

For the dimension, notice that the polynomials

$$P_k(X)=(X-a)(X-b)(X-c)X^k,\quad k\in\Bbb N$$ belong to $\mathcal U$. What we can conclude?