-3n(3n+6)+8=-82

Solution for -3n(3n+6)+8=-82 equation:

-3n(3n+6)+8=-82

We move all terms to the left:

-3n(3n+6)+8-(-82)=0

We add all the numbers together, and all the variables

-3n(3n+6)+90=0

We multiply parentheses

-9n^2-18n+90=0

a = -9; b = -18; c = +90;
Δ = b2-4ac
Δ = -182-4·(-9)·90
Δ = 3564
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3564}=\sqrt{324*11}=\sqrt{324}*\sqrt{11}=18\sqrt{11}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-18\sqrt{11}}{2*-9}=\frac{18-18\sqrt{11}}{-18}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+18\sqrt{11}}{2*-9}=\frac{18+18\sqrt{11}}{-18}
$