Complex Numbers and Quadratic Equations: JEE Main 2022 Mathematics question

Let $\alpha$, $\beta$ be the roots of the equation ${x^2} - 4\lambda x + 5 = 0$ and $\alpha$, $\gamma$ be the roots of the equation ${x^2} - \left( {3\sqrt 2 + 2\sqrt 3 } \right)x + 7 + 3\lambda \sqrt 3 = 0$, $\lambda$ > 0. If $\beta + \gamma = 3\sqrt 2$, then ${(\alpha + 2\beta + \gamma )^2}$ is equal to __________.