Quadratic Equation And Inequalities

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 __________.