(2x-3)(3+2)-6(x-2)(x+3)=-3

Solution for (2x-3)(3+2)-6(x-2)(x+3)=-3 equation:

(2x-3)(3+2)-6(x-2)(x+3)=-3

We move all terms to the left:

(2x-3)(3+2)-6(x-2)(x+3)-(-3)=0

We add all the numbers together, and all the variables

(2x-3)5-6(x-2)(x+3)-(-3)=0

We add all the numbers together, and all the variables

(2x-3)5-6(x-2)(x+3)+3=0

We multiply parentheses

10x-6(x-2)(x+3)-15+3=0

We multiply parentheses ..

-6(+x^2+3x-2x-6)+10x-15+3=0

We add all the numbers together, and all the variables

-6(+x^2+3x-2x-6)+10x-12=0

We multiply parentheses

-6x^2-18x+12x+10x+36-12=0

We add all the numbers together, and all the variables

-6x^2+4x+24=0

a = -6; b = 4; c = +24;
Δ = b2-4ac
Δ = 42-4·(-6)·24
Δ = 592
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:

$\sqrt{\Delta}=\sqrt{592}=\sqrt{16*37}=\sqrt{16}*\sqrt{37}=4\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-4\sqrt{37}}{2*-6}=\frac{-4-4\sqrt{37}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+4\sqrt{37}}{2*-6}=\frac{-4+4\sqrt{37}}{-12}
$