Symmetry

$\displaystyle \begin{aligned} E_1 \ : \ x^4y^5 + y^4x^5 = 810 \\ E_2 \ : \ x^3y^6 + y^3x^6 = 945 \end{aligned}$

$\begin{aligned} \displaystyle 3x^4y^4(x + y) + x^3y^3(x^3 + y^3) & = 3 \cdot 810 + 945 & \small \color{#3D99F6} \text{Add} \ 3 \cdot E_1 \ \text{to} \ E_2 \ \text{and factor} \\\\ (xy)^3(x^3 + 3x^2y + 3xy^2 + y^3) & = 3375 \\\\ (xy)^3(x + y)^3 & = 3375 & \small \color{#3D99F6} \text{Take the cube root of both sides} \\\\ xy(x + y) & = 15 & \small \color{#3D99F6} \text{Note that} \ E_1 = (xy)^3 \cdot xy(x + y) = 810 \\\\ \implies (xy)^3 & = \frac {810}{15} = 54 & \small \color{#3D99F6} \text{Note that} \ E_2 = (xy)^3(x^3 + y^3) = 945 \\\\ \implies x^3 + y^3 & = \frac{945}{54} = \frac{35}{2} \end{aligned}$

$\implies 2x^3 + (xy)^3 + 2y^3 = 54 + \frac{35}{2} \cdot 2 = 89$