1(5)2(x-3)=4(x)1(x-5)

Solution for 1(5)2(x-3)=4(x)1(x-5) equation:

Simplifying
1(5) * 2(x + -3) = 4(x) * 1(x + -5)

Reorder the terms:
1 * 5 * 2(-3 + x) = 4(x) * 1(x + -5)

Multiply 1 * 5
5 * 2(-3 + x) = 4(x) * 1(x + -5)

Multiply 5 * 2
10(-3 + x) = 4(x) * 1(x + -5)
(-3 * 10 + x * 10) = 4(x) * 1(x + -5)
(-30 + 10x) = 4(x) * 1(x + -5)

Reorder the terms:
-30 + 10x = 4x * 1(-5 + x)

Reorder the terms for easier multiplication:
-30 + 10x = 4 * 1x(-5 + x)

Multiply 4 * 1
-30 + 10x = 4x(-5 + x)
-30 + 10x = (-5 * 4x + x * 4x)
-30 + 10x = (-20x + 4x2)

Solving
-30 + 10x = -20x + 4x2

Solving for variable 'x'.

Combine like terms: 10x + 20x = 30x
-30 + 30x + -4x2 = -20x + 4x2 + 20x + -4x2

Reorder the terms:
-30 + 30x + -4x2 = -20x + 20x + 4x2 + -4x2

Combine like terms: -20x + 20x = 0
-30 + 30x + -4x2 = 0 + 4x2 + -4x2
-30 + 30x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-30 + 30x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-15 + 15x + -2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-15 + 15x + -2x2)' equal to zero and attempt to solve:

Simplifying
-15 + 15x + -2x2 = 0

Solving
-15 + 15x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
7.5 + -7.5x + x2 = 0

Move the constant term to the right:

Add '-7.5' to each side of the equation.
7.5 + -7.5x + -7.5 + x2 = 0 + -7.5

Reorder the terms:
7.5 + -7.5 + -7.5x + x2 = 0 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + -7.5x + x2 = 0 + -7.5
-7.5x + x2 = 0 + -7.5

Combine like terms: 0 + -7.5 = -7.5
-7.5x + x2 = -7.5

The x term is -7.5x.  Take half its coefficient (-3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
-7.5x + 14.0625 + x2 = -7.5 + 14.0625

Reorder the terms:
14.0625 + -7.5x + x2 = -7.5 + 14.0625

Combine like terms: -7.5 + 14.0625 = 6.5625
14.0625 + -7.5x + x2 = 6.5625

Factor a perfect square on the left side:
(x + -3.75)(x + -3.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(x + -3.75) equal to 2.561737691 and -2.561737691.

Subproblem 1
x + -3.75 = 2.561737691

Simplifying
x + -3.75 = 2.561737691

Reorder the terms:
-3.75 + x = 2.561737691

Solving
-3.75 + x = 2.561737691

Solving for variable 'x'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + x = 2.561737691 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + x = 2.561737691 + 3.75
x = 2.561737691 + 3.75

Combine like terms: 2.561737691 + 3.75 = 6.311737691
x = 6.311737691

Simplifying
x = 6.311737691

Subproblem 2
x + -3.75 = -2.561737691

Simplifying
x + -3.75 = -2.561737691

Reorder the terms:
-3.75 + x = -2.561737691

Solving
-3.75 + x = -2.561737691

Solving for variable 'x'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + x = -2.561737691 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + x = -2.561737691 + 3.75
x = -2.561737691 + 3.75

Combine like terms: -2.561737691 + 3.75 = 1.188262309
x = 1.188262309

Simplifying
x = 1.188262309

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.311737691, 1.188262309}
Solution
x = {6.311737691, 1.188262309}