12z(z+4)-4(z-4)=3(z-1)+4(z-2)

Solution for 12z(z+4)-4(z-4)=3(z-1)+4(z-2) equation:

Simplifying
12z(z + 4) + -4(z + -4) = 3(z + -1) + 4(z + -2)

Reorder the terms:
12z(4 + z) + -4(z + -4) = 3(z + -1) + 4(z + -2)
(4 * 12z + z * 12z) + -4(z + -4) = 3(z + -1) + 4(z + -2)
(48z + 12z2) + -4(z + -4) = 3(z + -1) + 4(z + -2)

Reorder the terms:
48z + 12z2 + -4(-4 + z) = 3(z + -1) + 4(z + -2)
48z + 12z2 + (-4 * -4 + z * -4) = 3(z + -1) + 4(z + -2)
48z + 12z2 + (16 + -4z) = 3(z + -1) + 4(z + -2)

Reorder the terms:
16 + 48z + -4z + 12z2 = 3(z + -1) + 4(z + -2)

Combine like terms: 48z + -4z = 44z
16 + 44z + 12z2 = 3(z + -1) + 4(z + -2)

Reorder the terms:
16 + 44z + 12z2 = 3(-1 + z) + 4(z + -2)
16 + 44z + 12z2 = (-1 * 3 + z * 3) + 4(z + -2)
16 + 44z + 12z2 = (-3 + 3z) + 4(z + -2)

Reorder the terms:
16 + 44z + 12z2 = -3 + 3z + 4(-2 + z)
16 + 44z + 12z2 = -3 + 3z + (-2 * 4 + z * 4)
16 + 44z + 12z2 = -3 + 3z + (-8 + 4z)

Reorder the terms:
16 + 44z + 12z2 = -3 + -8 + 3z + 4z

Combine like terms: -3 + -8 = -11
16 + 44z + 12z2 = -11 + 3z + 4z

Combine like terms: 3z + 4z = 7z
16 + 44z + 12z2 = -11 + 7z

Solving
16 + 44z + 12z2 = -11 + 7z

Solving for variable 'z'.

Reorder the terms:
16 + 11 + 44z + -7z + 12z2 = -11 + 7z + 11 + -7z

Combine like terms: 16 + 11 = 27
27 + 44z + -7z + 12z2 = -11 + 7z + 11 + -7z

Combine like terms: 44z + -7z = 37z
27 + 37z + 12z2 = -11 + 7z + 11 + -7z

Reorder the terms:
27 + 37z + 12z2 = -11 + 11 + 7z + -7z

Combine like terms: -11 + 11 = 0
27 + 37z + 12z2 = 0 + 7z + -7z
27 + 37z + 12z2 = 7z + -7z

Combine like terms: 7z + -7z = 0
27 + 37z + 12z2 = 0

Begin completing the square.  Divide all terms by
12 the coefficient of the squared term: 

Divide each side by '12'.
2.25 + 3.083333333z + z2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + 3.083333333z + -2.25 + z2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + 3.083333333z + z2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + 3.083333333z + z2 = 0 + -2.25
3.083333333z + z2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
3.083333333z + z2 = -2.25

The z term is 3.083333333z.  Take half its coefficient (1.541666667).
Square it (2.376736112) and add it to both sides.

Add '2.376736112' to each side of the equation.
3.083333333z + 2.376736112 + z2 = -2.25 + 2.376736112

Reorder the terms:
2.376736112 + 3.083333333z + z2 = -2.25 + 2.376736112

Combine like terms: -2.25 + 2.376736112 = 0.126736112
2.376736112 + 3.083333333z + z2 = 0.126736112

Factor a perfect square on the left side:
(z + 1.541666667)(z + 1.541666667) = 0.126736112

Calculate the square root of the right side: 0.356000157

Break this problem into two subproblems by setting 
(z + 1.541666667) equal to 0.356000157 and -0.356000157.

Subproblem 1
z + 1.541666667 = 0.356000157

Simplifying
z + 1.541666667 = 0.356000157

Reorder the terms:
1.541666667 + z = 0.356000157

Solving
1.541666667 + z = 0.356000157

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-1.541666667' to each side of the equation.
1.541666667 + -1.541666667 + z = 0.356000157 + -1.541666667

Combine like terms: 1.541666667 + -1.541666667 = 0.000000000
0.000000000 + z = 0.356000157 + -1.541666667
z = 0.356000157 + -1.541666667

Combine like terms: 0.356000157 + -1.541666667 = -1.18566651
z = -1.18566651

Simplifying
z = -1.18566651

Subproblem 2
z + 1.541666667 = -0.356000157

Simplifying
z + 1.541666667 = -0.356000157

Reorder the terms:
1.541666667 + z = -0.356000157

Solving
1.541666667 + z = -0.356000157

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-1.541666667' to each side of the equation.
1.541666667 + -1.541666667 + z = -0.356000157 + -1.541666667

Combine like terms: 1.541666667 + -1.541666667 = 0.000000000
0.000000000 + z = -0.356000157 + -1.541666667
z = -0.356000157 + -1.541666667

Combine like terms: -0.356000157 + -1.541666667 = -1.897666824
z = -1.897666824

Simplifying
z = -1.897666824

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-1.18566651, -1.897666824}