If the system of equations
$$2x + y - z = 5$$
$$2x - 5y + \lambda z = \mu$$
$$x + 2y - 5z = 7$$
has infinitely many solutions, then $$(\lambda + \mu)^2 + (\lambda - \mu)^2$$ is equal to

For infinitely many solutions, $$D = D_1 = D_2 = D_3 = 0$$.

Coefficient matrix determinant:

$$D = \begin{vmatrix} 2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5 \end{vmatrix}$$

$$= 2(25-2\lambda) - 1(-10-\lambda) + (-1)(4+5)$$

$$= 50 - 4\lambda + 10 + \lambda - 9 = 51 - 3\lambda$$

For $$D = 0$$: $$\lambda = 17$$

Now find $$\mu$$ using $$D_1 = 0$$:

$$D_1 = \begin{vmatrix} 5 & 1 & -1 \\ \mu & -5 & 17 \\ 7 & 2 & -5 \end{vmatrix}$$

$$= 5(25-34) - 1(-5\mu-119) + (-1)(2\mu+35)$$

$$= 5(-9) + 5\mu + 119 - 2\mu - 35$$

$$= -45 + 3\mu + 84 = 3\mu + 39$$

$$D_1 = 0$$: $$\mu = -13$$

$$(\lambda + \mu)^2 + (\lambda - \mu)^2 = (17-13)^2 + (17+13)^2 = 16 + 900 = 916$$

This matches option 2: 916.