# Determine a polynomium $q \in P_3(\mathbb{R})$ where the coordinat vector $[q]_v$ is equal to v where v is a vector

I have to determine a polynomium $$q \in P_3(\mathbb{R})$$ where the coordinat vector $$[q]_v$$ is equal to v where v is a vector given by $$v = \begin{pmatrix} \alpha \beta \\ -\alpha -\beta \\ 1 \end{pmatrix}$$ My books has a note saying that the coordinat vector $$[q]_v$$ is given by the vector $$\begin{pmatrix} a_1 \\ a_2 \\ ... \\ a_n \end{pmatrix} \in P_3(\mathbb{R})$$ where $$v = a_1 \cdot q_1 + a_2 \cdot q_2 + a_3 \cdot q_3$$ where $$Q = (q_1,q_2,q_3)$$ is a basis given by $$Q = (1,X,X^2)$$. I am not really sure that I am explaining this correct so I have this is correct so far.

In regards I have then said that the polynomium q will be $$q = \frac{1}{X^2} + \frac{-\alpha -\beta}{X} + \alpha \beta$$ Is this correct? Or am I doing it completely wrong?

$$q=\alpha \beta q_1-(\alpha + \beta)q_2+q_3,$$
$$q(X)=\alpha \beta -(\alpha + \beta) X+X^2.$$