Let α, β be the roots of the equation x^2- a x- b =0 with Im (α)< Im (β). Let P_n =α^n -β^n. If P_3=-5 √ 7 i, P_4=-3 √ 7 i, P_5=11 √ 7 i and P_6=45 √…

Let $\displaystyle \alpha, \beta$ be the roots of the equation $\displaystyle x^2-\mathrm{a} x-\mathrm{b}=0$ with $\displaystyle \operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $\displaystyle \mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$. If $\displaystyle \mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i$ and $\displaystyle \mathrm{P}_6=45 \sqrt{7} i$, then $\displaystyle \left|\alpha^4+\beta^4\right|$ is equal to $\displaystyle \_\_\_\_$.