(3+z)(4z+2)=0

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

Simplifying
(3 + z)(4z + 2) = 0

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

Multiply (3 + z) * (2 + 4z)
(3(2 + 4z) + z(2 + 4z)) = 0
((2 * 3 + 4z * 3) + z(2 + 4z)) = 0
((6 + 12z) + z(2 + 4z)) = 0
(6 + 12z + (2 * z + 4z * z)) = 0
(6 + 12z + (2z + 4z2)) = 0

Combine like terms: 12z + 2z = 14z
(6 + 14z + 4z2) = 0

Solving
6 + 14z + 4z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 7z + 2z2) = 0

Factor a trinomial.
2((3 + z)(1 + 2z)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + z)' equal to zero and attempt to solve:

Simplifying
3 + z = 0

Solving
3 + z = 0

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

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

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

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

Simplifying
z = -3
Subproblem 2
Set the factor '(1 + 2z)' equal to zero and attempt to solve:

Simplifying
1 + 2z = 0

Solving
1 + 2z = 0

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

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

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

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

Divide each side by '2'.
z = -0.5

Simplifying
z = -0.5
Solution
z = {-3, -0.5}