Proving Theorem: subspace of polynomials of degree two or less?

Question

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less?

I know I need to show that $a+b+c=0$ when $(ax^2+bx+c)$ but I am not sure where to go with this proof

Answer 1

Hint. See if you can fill in the gaps in the following proof.

1. The zero polynomial in $P_2$ is $0+0x+0x^2$. This is in $S$ because ...
2. Let $p=a+bx+cx^2$ and $q=d+ex+fx^2$ be polynomials in $S$.
   This means that $a+b+c=0$ and ...
   Now adding the two polynomials and simplifying gives $$p+q=\cdots\ .$$
   The sum of the coefficients of $p+q$ is ... and so $p+q$ is in $S$.
   Therefore $S$ is closed under addition.
3. ... (Something similar for scalar multiplication.)