4x=-x^2-2

Solution for 4x=-x^2-2 equation:

4x=-x^2-2

We move all terms to the left:

4x-(-x^2-2)=0

We get rid of parentheses

x^2+4x+2=0

a = 1; b = 4; c = +2;
Δ = b2-4ac
Δ = 42-4·1·2
Δ = 8
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{8}=\sqrt{4*2}=\sqrt{4}*\sqrt{2}=2\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{2}}{2*1}=\frac{-4-2\sqrt{2}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{2}}{2*1}=\frac{-4+2\sqrt{2}}{2}
$