z[4(z+2)-5(z-1)]=3(z-1)(z+2)

Solution for z[4(z+2)-5(z-1)]=3(z-1)(z+2) equation:

Simplifying
z[4(z + 2) + -5(z + -1)] = 3(z + -1)(z + 2)

Reorder the terms:
z[4(2 + z) + -5(z + -1)] = 3(z + -1)(z + 2)
z[(2 * 4 + z * 4) + -5(z + -1)] = 3(z + -1)(z + 2)
z[(8 + 4z) + -5(z + -1)] = 3(z + -1)(z + 2)

Reorder the terms:
z[8 + 4z + -5(-1 + z)] = 3(z + -1)(z + 2)
z[8 + 4z + (-1 * -5 + z * -5)] = 3(z + -1)(z + 2)
z[8 + 4z + (5 + -5z)] = 3(z + -1)(z + 2)

Reorder the terms:
z[8 + 5 + 4z + -5z] = 3(z + -1)(z + 2)

Combine like terms: 8 + 5 = 13
z[13 + 4z + -5z] = 3(z + -1)(z + 2)

Combine like terms: 4z + -5z = -1z
z[13 + -1z] = 3(z + -1)(z + 2)
[13 * z + -1z * z] = 3(z + -1)(z + 2)
[13z + -1z2] = 3(z + -1)(z + 2)

Reorder the terms:
13z + -1z2 = 3(-1 + z)(z + 2)

Reorder the terms:
13z + -1z2 = 3(-1 + z)(2 + z)

Multiply (-1 + z) * (2 + z)
13z + -1z2 = 3(-1(2 + z) + z(2 + z))
13z + -1z2 = 3((2 * -1 + z * -1) + z(2 + z))
13z + -1z2 = 3((-2 + -1z) + z(2 + z))
13z + -1z2 = 3(-2 + -1z + (2 * z + z * z))
13z + -1z2 = 3(-2 + -1z + (2z + z2))

Combine like terms: -1z + 2z = 1z
13z + -1z2 = 3(-2 + 1z + z2)
13z + -1z2 = (-2 * 3 + 1z * 3 + z2 * 3)
13z + -1z2 = (-6 + 3z + 3z2)

Solving
13z + -1z2 = -6 + 3z + 3z2

Solving for variable 'z'.

Reorder the terms:
6 + 13z + -3z + -1z2 + -3z2 = -6 + 3z + 3z2 + 6 + -3z + -3z2

Combine like terms: 13z + -3z = 10z
6 + 10z + -1z2 + -3z2 = -6 + 3z + 3z2 + 6 + -3z + -3z2

Combine like terms: -1z2 + -3z2 = -4z2
6 + 10z + -4z2 = -6 + 3z + 3z2 + 6 + -3z + -3z2

Reorder the terms:
6 + 10z + -4z2 = -6 + 6 + 3z + -3z + 3z2 + -3z2

Combine like terms: -6 + 6 = 0
6 + 10z + -4z2 = 0 + 3z + -3z + 3z2 + -3z2
6 + 10z + -4z2 = 3z + -3z + 3z2 + -3z2

Combine like terms: 3z + -3z = 0
6 + 10z + -4z2 = 0 + 3z2 + -3z2
6 + 10z + -4z2 = 3z2 + -3z2

Combine like terms: 3z2 + -3z2 = 0
6 + 10z + -4z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 5z + -2z2) = 0

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

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

Simplifying
3 + -1z = 0

Solving
3 + -1z = 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 + -1z = 0 + -3

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

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

Divide each side by '-1'.
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}