x²+(x+2)²=10² equation

Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$x^{2} + \left(x + 2\right)^{2} = 10^{2}$$
to
$$\left(x^{2} + \left(x + 2\right)^{2}\right) - 10^{2} = 0$$
Expand the expression in the equation
$$\left(x^{2} + \left(x + 2\right)^{2}\right) - 10^{2} = 0$$
We get the quadratic equation
$$2 x^{2} + 4 x - 100 + 4 = 0$$
This equation is of the form
$$a\ x^2 + b\ x + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = 4$$
$$c = -96$$
, then
$$D = b^2 - 4\ a\ c = $$
$$4^{2} - 2 \cdot 4 \left(-96\right) = 784$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 6$$
Simplify
$$x_{2} = -8$$
Simplify