Let α, β be the roots of the equation x ^ 2 - 4 + 5 = 0 and α, γ be...

Given $\alpha \text{ and } \beta$ are roots of ${x}^{2} - 4\lambdax + 5 = 0$

So, $\alpha + \beta = 4\lambda$ and $\alpha\beta = 5$ .......(i)

And $\alpha \text{ and } \gamma$ are roots of ${x}^{2} - \left(3\sqrt{2} + 2\sqrt{3}\right)x + \left(7 + 3\lambda\sqrt{3}\right) = 0$

So, $\alpha + y = 3\sqrt{2} + 2\sqrt{3}$ and $\alpha\gamma = 7 + 3\lambda\sqrt{3}$ .........(ii)

Given, if $\beta + \gamma = 3\sqrt{2}$

So, from equation (i) and (ii) we get, $\alpha = 2\lambda + \sqrt{3}$ and

$\beta = 2\lambda - \sqrt{3}$, Now by product of roots we get, $4{\lambda}^{2} - 3 = 5 ⇒ \lambda = \sqrt{2}$

$∴ {\left(\alpha + 2\beta + \lambda\right)}^{2} = {\left(\alpha + \beta + \beta + \lambda\right)}^{2} = {\left(4\lambda + 3\sqrt{2}\right)}^{2}$

$= {\left(4\sqrt{2} + 3\sqrt{2}\right)}^{2} = {\left(7\sqrt{2}\right)}^{2} = 98$