(3z^2+5z)+(z^2+2z)+(z-4)=0

Solution for (3z^2+5z)+(z^2+2z)+(z-4)=0 equation:

Simplifying
(3z2 + 5z) + (z2 + 2z) + (z + -4) = 0

Reorder the terms:
(5z + 3z2) + (z2 + 2z) + (z + -4) = 0

Remove parenthesis around (5z + 3z2)
5z + 3z2 + (z2 + 2z) + (z + -4) = 0

Reorder the terms:
5z + 3z2 + (2z + z2) + (z + -4) = 0

Remove parenthesis around (2z + z2)
5z + 3z2 + 2z + z2 + (z + -4) = 0

Reorder the terms:
5z + 3z2 + 2z + z2 + (-4 + z) = 0

Remove parenthesis around (-4 + z)
5z + 3z2 + 2z + z2 + -4 + z = 0

Reorder the terms:
-4 + 5z + 2z + z + 3z2 + z2 = 0

Combine like terms: 5z + 2z = 7z
-4 + 7z + z + 3z2 + z2 = 0

Combine like terms: 7z + z = 8z
-4 + 8z + 3z2 + z2 = 0

Combine like terms: 3z2 + z2 = 4z2
-4 + 8z + 4z2 = 0

Solving
-4 + 8z + 4z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-1 + 2z + z2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-1 + 2z + z2)' equal to zero and attempt to solve:

Simplifying
-1 + 2z + z2 = 0

Solving
-1 + 2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 2z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2z + z2 = 0 + 1
2z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
2z + z2 = 1

The z term is 2z.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z + 1 + z2 = 1 + 1

Reorder the terms:
1 + 2z + z2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + 2z + z2 = 2

Factor a perfect square on the left side:
(z + 1)(z + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
z + 1 = 1.414213562

Simplifying
z + 1 = 1.414213562

Reorder the terms:
1 + z = 1.414213562

Solving
1 + z = 1.414213562

Solving for variable 'z'.

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

Add '-1' to each side of the equation.
1 + -1 + z = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + z = 1.414213562 + -1
z = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
z = 0.414213562

Simplifying
z = 0.414213562

Subproblem 2
z + 1 = -1.414213562

Simplifying
z + 1 = -1.414213562

Reorder the terms:
1 + z = -1.414213562

Solving
1 + z = -1.414213562

Solving for variable 'z'.

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

Add '-1' to each side of the equation.
1 + -1 + z = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + z = -1.414213562 + -1
z = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
z = -2.414213562

Simplifying
z = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.414213562, -2.414213562}
Solution
z = {0.414213562, -2.414213562}