(z+1)(z+4)+3(z-2)(z-1)=4z(z-6)

Solution for (z+1)(z+4)+3(z-2)(z-1)=4z(z-6) equation:

Simplifying
(z + 1)(z + 4) + 3(z + -2)(z + -1) = 4z(z + -6)

Reorder the terms:
(1 + z)(z + 4) + 3(z + -2)(z + -1) = 4z(z + -6)

Reorder the terms:
(1 + z)(4 + z) + 3(z + -2)(z + -1) = 4z(z + -6)

Multiply (1 + z) * (4 + z)
(1(4 + z) + z(4 + z)) + 3(z + -2)(z + -1) = 4z(z + -6)
((4 * 1 + z * 1) + z(4 + z)) + 3(z + -2)(z + -1) = 4z(z + -6)
((4 + 1z) + z(4 + z)) + 3(z + -2)(z + -1) = 4z(z + -6)
(4 + 1z + (4 * z + z * z)) + 3(z + -2)(z + -1) = 4z(z + -6)
(4 + 1z + (4z + z2)) + 3(z + -2)(z + -1) = 4z(z + -6)

Combine like terms: 1z + 4z = 5z
(4 + 5z + z2) + 3(z + -2)(z + -1) = 4z(z + -6)

Reorder the terms:
4 + 5z + z2 + 3(-2 + z)(z + -1) = 4z(z + -6)

Reorder the terms:
4 + 5z + z2 + 3(-2 + z)(-1 + z) = 4z(z + -6)

Multiply (-2 + z) * (-1 + z)
4 + 5z + z2 + 3(-2(-1 + z) + z(-1 + z)) = 4z(z + -6)
4 + 5z + z2 + 3((-1 * -2 + z * -2) + z(-1 + z)) = 4z(z + -6)
4 + 5z + z2 + 3((2 + -2z) + z(-1 + z)) = 4z(z + -6)
4 + 5z + z2 + 3(2 + -2z + (-1 * z + z * z)) = 4z(z + -6)
4 + 5z + z2 + 3(2 + -2z + (-1z + z2)) = 4z(z + -6)

Combine like terms: -2z + -1z = -3z
4 + 5z + z2 + 3(2 + -3z + z2) = 4z(z + -6)
4 + 5z + z2 + (2 * 3 + -3z * 3 + z2 * 3) = 4z(z + -6)
4 + 5z + z2 + (6 + -9z + 3z2) = 4z(z + -6)

Reorder the terms:
4 + 6 + 5z + -9z + z2 + 3z2 = 4z(z + -6)

Combine like terms: 4 + 6 = 10
10 + 5z + -9z + z2 + 3z2 = 4z(z + -6)

Combine like terms: 5z + -9z = -4z
10 + -4z + z2 + 3z2 = 4z(z + -6)

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

Reorder the terms:
10 + -4z + 4z2 = 4z(-6 + z)
10 + -4z + 4z2 = (-6 * 4z + z * 4z)
10 + -4z + 4z2 = (-24z + 4z2)

Add '-4z2' to each side of the equation.
10 + -4z + 4z2 + -4z2 = -24z + 4z2 + -4z2

Combine like terms: 4z2 + -4z2 = 0
10 + -4z + 0 = -24z + 4z2 + -4z2
10 + -4z = -24z + 4z2 + -4z2

Combine like terms: 4z2 + -4z2 = 0
10 + -4z = -24z + 0
10 + -4z = -24z

Solving
10 + -4z = -24z

Solving for variable 'z'.

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

Add '24z' to each side of the equation.
10 + -4z + 24z = -24z + 24z

Combine like terms: -4z + 24z = 20z
10 + 20z = -24z + 24z

Combine like terms: -24z + 24z = 0
10 + 20z = 0

Add '-10' to each side of the equation.
10 + -10 + 20z = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 20z = 0 + -10
20z = 0 + -10

Combine like terms: 0 + -10 = -10
20z = -10

Divide each side by '20'.
z = -0.5

Simplifying
z = -0.5