Let's first simplify the equation on both sides.

-2x(1-x) + 3(x-4) = -x(x+2) - 6x

Expanding the terms gives:

-2x + 2x^2 + 3x - 12 = -x^2 - 2x - 6x

Combining like terms:

2x^2 - 2x + 3x - 12 = -x^2 - 2x - 6x

This simplifies to:

2x^2 + x - 12 = -x^2 - 8x

Now, let's combine like terms and move all terms to one side of the equation:

2x^2 + x - 12 + x^2 + 8x = 0

3x^2 + 9x - 12 = 0

This is a quadratic equation, which can be solved using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 3, b = 9, and c = -12. Plugging into the formula:

x = (-9 ± sqrt(81 + 144)) / 6
x = (-9 ± sqrt(225)) / 6
x = (-9 ± 15) / 6

This gives two solutions:

x = (6/6) = 1
x = (-24/6) = -4

Therefore, the solutions to the equation are x = 1 and x = -4.