(4z+1)(4+z)=0

Solution for (4z+1)(4+z)=0 equation:

Simplifying
(4z + 1)(4 + z) = 0

Reorder the terms:
(1 + 4z)(4 + z) = 0

Multiply (1 + 4z) * (4 + z)
(1(4 + z) + 4z * (4 + z)) = 0
((4 * 1 + z * 1) + 4z * (4 + z)) = 0
((4 + 1z) + 4z * (4 + z)) = 0
(4 + 1z + (4 * 4z + z * 4z)) = 0
(4 + 1z + (16z + 4z2)) = 0

Combine like terms: 1z + 16z = 17z
(4 + 17z + 4z2) = 0

Solving
4 + 17z + 4z2 = 0

Solving for variable 'z'.

Factor a trinomial.
(4 + z)(1 + 4z) = 0

Subproblem 1
Set the factor '(4 + z)' equal to zero and attempt to solve:

Simplifying
4 + z = 0

Solving
4 + z = 0

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

Add '-4' to each side of the equation.
4 + -4 + z = 0 + -4

Combine like terms: 4 + -4 = 0
0 + z = 0 + -4
z = 0 + -4

Combine like terms: 0 + -4 = -4
z = -4

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

Simplifying
1 + 4z = 0

Solving
1 + 4z = 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 + 4z = 0 + -1

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

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

Divide each side by '4'.
z = -0.25

Simplifying
z = -0.25
Solution
z = {-4, -0.25}