$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$.
I need to show this function is an inner product:
$$\langle p,q\rangle=\sum_{j=0}^n a_j\overline{b}_j$$
Specifically I need to show it satifies the properties of an inner product: positivity, conjugate symmetry, homogeneity and linearity. I can't get past positivity.
Since the space $V$ contains polynomials of degree $\leq n$, we can do a really easy "cheat".
The map taking $a_0+a_1x+\cdots a_nx^n$ to $(a_0,a_1,\cdots,a_n)$ is a vector space isomorphism from $V$ to $\mathbb{C}^n$. Also, the inner product defined here then becomes the standard inner product on $\mathbb{C}^n$. So, you can now deduce that your "inner product" is really an inner product.
In any case, I would recommend you to go through the proof why the standard inner product on $\mathbb{C}^n$ satisfies the properties like positivity, conjugate symmetry, homogeneity and linearity. The same proof shall apply in this case.