Updated improved question:
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1). $$
(i) Prove that $\langle\cdot,\cdot\rangle$ defines an inner product.
Could we just do this $f(a)=0$ and $f'(a)=0$ then $f(x)$ is divisible by $(x-a)^2$ ?
If so how would we solve this?
Can someone please help me with this proof for part (i). It is frustrating me.