x+(x+1)+x(x+2)=66

Solution for x+(x+1)+x(x+2)=66 equation:

x+(x+1)+x(x+2)=66

We move all terms to the left:

x+(x+1)+x(x+2)-(66)=0

We multiply parentheses

x^2+x+(x+1)+2x-66=0

We get rid of parentheses

x^2+x+x+2x+1-66=0

We add all the numbers together, and all the variables

x^2+4x-65=0

a = 1; b = 4; c = -65;
Δ = b2-4ac
Δ = 42-4·1·(-65)
Δ = 276
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:

$\sqrt{\Delta}=\sqrt{276}=\sqrt{4*69}=\sqrt{4}*\sqrt{69}=2\sqrt{69}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{69}}{2*1}=\frac{-4-2\sqrt{69}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{69}}{2*1}=\frac{-4+2\sqrt{69}}{2}
$