Orthogonality in Space of Polynomials of Degree at Most 2

Question

Let $E$ be the space of polynomials of degree at most $2$. On $E$ define $\langle f,g \rangle := f(-1)\overline{g(-1)}+f(0)\overline{g(0)}+f(1)\overline{g(1)}$ for $f,g \in E$.

a). Show that this defines an inner product on $E$.

b). Describe $\{t^2-1\}^{\perp}$.

c). Show that the polynomials $t^2-1, t^2-t$ are orthogonal, and find a nonzero polynomial $p \in E$ that is orthogonal to both of them.

For the first part, I believe I was able to satisfy the 4 properties required as per my textbook:

1. Sesquilinear: $\langle \lambda f+ \mu g, h \rangle = \lambda \langle f,h \rangle + \mu \langle g, h \rangle$.
2. Symmetric: $\langle f,g \rangle = \overline{\langle g, f \rangle}, \quad f,g \in E$.
3. Positive: $\langle f, f \rangle \geq 0.$
4. Definite: $\langle f,f \rangle = 0 \Rightarrow f = 0$.

I didn't want to include the details of this in order to save some space on this post. What I am struggling on is parts (b) and (c).

For (b), I have that $\{t^2-1\}^{\perp} = \{f \in E: f \perp g \forall g \in \{t^2-1\} \}$. This would mean that the orthogonal compliment of $\{t^2-1\}$ is the set of all polynomials of degree at most 2 whose inner product with $g \in \{t^2-1\}$ is equal to 0. However, is this an accurate enough description or is it too general?

For (c), using the inner product defined on $E$, I have the following:

I rewrote the inner product as $\langle f,g \rangle = \sum_{i=-1}^{1} f(i)\overline{g(i)} = \left. t^2-t\overline{(t^2-1)}\right|_{-1} + \left. t^2-\overline{t(t^2-1)}\right|_0 + \left. t^2-t\overline{(t^2-1)}\right|_1$. Since I have to use the conjugate of $t^2-1$, then I change that term to $t^2+1$ and then evaluate. But I got a value of 4 which implies that $f=t^2-t$ and $g=t^2-1$ are not orthogonal. Was there something wrong with my calculation? As for the next part, I have not gotten there yet, but I assume that I would have to do some sort of long division first.

As always, any help/suggestions/advice would be greatly appreciated, thanks in advance. I am using the textbook Functional Analysis An Elementary Introduction by Markus Haase.