(n+3)(n+1)=52

Solution for (n+3)(n+1)=52 equation:

(n+3)(n+1)=52

We move all terms to the left:

(n+3)(n+1)-(52)=0

We multiply parentheses ..

(+n^2+n+3n+3)-52=0

We get rid of parentheses

n^2+n+3n+3-52=0

We add all the numbers together, and all the variables

n^2+4n-49=0

a = 1; b = 4; c = -49;
Δ = b2-4ac
Δ = 42-4·1·(-49)
Δ = 212
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:

$\sqrt{\Delta}=\sqrt{212}=\sqrt{4*53}=\sqrt{4}*\sqrt{53}=2\sqrt{53}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{53}}{2*1}=\frac{-4-2\sqrt{53}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{53}}{2*1}=\frac{-4+2\sqrt{53}}{2}
$