-5n(2n+3)+3(3n+4)=-8

Solution for -5n(2n+3)+3(3n+4)=-8 equation:

-5n(2n+3)+3(3n+4)=-8

We move all terms to the left:

-5n(2n+3)+3(3n+4)-(-8)=0

We add all the numbers together, and all the variables

-5n(2n+3)+3(3n+4)+8=0

We multiply parentheses

-10n^2-15n+9n+12+8=0

We add all the numbers together, and all the variables

-10n^2-6n+20=0

a = -10; b = -6; c = +20;
Δ = b2-4ac
Δ = -62-4·(-10)·20
Δ = 836
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:

$\sqrt{\Delta}=\sqrt{836}=\sqrt{4*209}=\sqrt{4}*\sqrt{209}=2\sqrt{209}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{209}}{2*-10}=\frac{6-2\sqrt{209}}{-20}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{209}}{2*-10}=\frac{6+2\sqrt{209}}{-20}
$