There're a few more that needs to get done...

I don't create this problem. All of the credits go to my teacher for writing this problem.

The original problem was:

Solve the following system of equations.

$\large \left\{ \begin{aligned} \sqrt{x + 2}(x - y + 3) &= \sqrt y\\ x^2 + (x + 3)(2x - y + 5) &= x + 16 \end{aligned} \right.$

There are two real solutions to the system of equations, $(x, y) \in \{(x_1, y_1), (x_2, y_2)\}$ such that $5(x_1 + y_1) + 2(x_2 + y_2) = 0$ .

Calculate the value of $x_1y_2 + x_2y_1 + y_1y_2$ .