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

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

Simplifying
3z(z + 4) = -1(z2 + -4)

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

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

Solving
12z + 3z2 = 4 + -1z2

Solving for variable 'z'.

Reorder the terms:
-4 + 12z + 3z2 + z2 = 4 + -1z2 + -4 + z2

Combine like terms: 3z2 + z2 = 4z2
-4 + 12z + 4z2 = 4 + -1z2 + -4 + z2

Reorder the terms:
-4 + 12z + 4z2 = 4 + -4 + -1z2 + z2

Combine like terms: 4 + -4 = 0
-4 + 12z + 4z2 = 0 + -1z2 + z2
-4 + 12z + 4z2 = -1z2 + z2

Combine like terms: -1z2 + z2 = 0
-4 + 12z + 4z2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-1 + 3z + z2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-1 + 3z + z2)' equal to zero and attempt to solve:

Simplifying
-1 + 3z + z2 = 0

Solving
-1 + 3z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 3z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3z + z2 = 0 + 1
3z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
3z + z2 = 1

The z term is 3z.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z + 2.25 + z2 = 1 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + 3z + z2 = 3.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + 1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
z + 1.5 = 1.802775638

Simplifying
z + 1.5 = 1.802775638

Reorder the terms:
1.5 + z = 1.802775638

Solving
1.5 + z = 1.802775638

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = 1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = 1.802775638 + -1.5
z = 1.802775638 + -1.5

Combine like terms: 1.802775638 + -1.5 = 0.302775638
z = 0.302775638

Simplifying
z = 0.302775638

Subproblem 2
z + 1.5 = -1.802775638

Simplifying
z + 1.5 = -1.802775638

Reorder the terms:
1.5 + z = -1.802775638

Solving
1.5 + z = -1.802775638

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z = -1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = -1.802775638 + -1.5
z = -1.802775638 + -1.5

Combine like terms: -1.802775638 + -1.5 = -3.302775638
z = -3.302775638

Simplifying
z = -3.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.302775638, -3.302775638}
Solution
z = {0.302775638, -3.302775638}