Is there a closed form for the polynomials satisfying this recurrence relation?

Let $\alpha,\beta$ be the solutions of $t^2=xt+1$.

Since $$\alpha+\beta=x,\quad \alpha\beta=-1$$ we have $$P(n+1)=(\alpha+\beta)P(n)-\alpha\beta P(n-1)$$

So, we get $$P(n+1)-\alpha P(n)=\beta (P(n)-\alpha P(n-1))=\cdots =\beta^{n-1}(P(2)-\alpha P(1))$$ and $$P(n+1)-\beta P(n)=\alpha (P(n)-\beta P(n-1))=\cdots =\alpha^{n-1}(P(2)-\beta P(1))$$

Subtracting the latter from the former gives $$(\beta-\alpha)P(n)=\beta^{n-1}(x-\alpha)-\alpha^{n-1}(x-\beta),$$ i.e. $$P(n)=\frac{\beta^{n-1}(x-\alpha)-\alpha^{n-1}(x-\beta)}{\beta-\alpha}$$ where $$\alpha=\frac{x-\sqrt{x^2+4}}{2},\quad\beta=\frac{x+\sqrt{x^2+4}}{2}$$

Is there a closed form for the polynomials satisfying this recurrence relation?

Hint. The answer is yes. One may use the characteristic equation (see here or here) to get $$ P(n)=c_1 \left(\frac{\sqrt{x^2+4}+x}2\right)^n+c_2 \left(\frac{x-\sqrt{x^2+4}}2\right)^n. $$