(4n+5)(8n-5)=180

Solution for (4n+5)(8n-5)=180 equation:

(4n+5)(8n-5)=180

We move all terms to the left:

(4n+5)(8n-5)-(180)=0

We multiply parentheses ..

(+32n^2-20n+40n-25)-180=0

We get rid of parentheses

32n^2-20n+40n-25-180=0

We add all the numbers together, and all the variables

32n^2+20n-205=0

a = 32; b = 20; c = -205;
Δ = b2-4ac
Δ = 202-4·32·(-205)
Δ = 26640
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:

$\sqrt{\Delta}=\sqrt{26640}=\sqrt{144*185}=\sqrt{144}*\sqrt{185}=12\sqrt{185}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-12\sqrt{185}}{2*32}=\frac{-20-12\sqrt{185}}{64}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+12\sqrt{185}}{2*32}=\frac{-20+12\sqrt{185}}{64}
$