(3x+17)+x(2x-10)+x+71=180

Solution for (3x+17)+x(2x-10)+x+71=180 equation:

Simplifying
(3x + 17) + x(2x + -10) + x + 71 = 180

Reorder the terms:
(17 + 3x) + x(2x + -10) + x + 71 = 180

Remove parenthesis around (17 + 3x)
17 + 3x + x(2x + -10) + x + 71 = 180

Reorder the terms:
17 + 3x + x(-10 + 2x) + x + 71 = 180
17 + 3x + (-10 * x + 2x * x) + x + 71 = 180
17 + 3x + (-10x + 2x2) + x + 71 = 180

Reorder the terms:
17 + 71 + 3x + -10x + x + 2x2 = 180

Combine like terms: 17 + 71 = 88
88 + 3x + -10x + x + 2x2 = 180

Combine like terms: 3x + -10x = -7x
88 + -7x + x + 2x2 = 180

Combine like terms: -7x + x = -6x
88 + -6x + 2x2 = 180

Solving
88 + -6x + 2x2 = 180

Solving for variable 'x'.

Reorder the terms:
88 + -180 + -6x + 2x2 = 180 + -180

Combine like terms: 88 + -180 = -92
-92 + -6x + 2x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-92 + -6x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-46 + -3x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-46 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
-46 + -3x + x2 = 0

Solving
-46 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '46' to each side of the equation.
-46 + -3x + 46 + x2 = 0 + 46

Reorder the terms:
-46 + 46 + -3x + x2 = 0 + 46

Combine like terms: -46 + 46 = 0
0 + -3x + x2 = 0 + 46
-3x + x2 = 0 + 46

Combine like terms: 0 + 46 = 46
-3x + x2 = 46

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 46 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 46 + 2.25

Combine like terms: 46 + 2.25 = 48.25
2.25 + -3x + x2 = 48.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 48.25

Calculate the square root of the right side: 6.946221995

Break this problem into two subproblems by setting 
(x + -1.5) equal to 6.946221995 and -6.946221995.

Subproblem 1
x + -1.5 = 6.946221995

Simplifying
x + -1.5 = 6.946221995

Reorder the terms:
-1.5 + x = 6.946221995

Solving
-1.5 + x = 6.946221995

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 6.946221995 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 6.946221995 + 1.5
x = 6.946221995 + 1.5

Combine like terms: 6.946221995 + 1.5 = 8.446221995
x = 8.446221995

Simplifying
x = 8.446221995

Subproblem 2
x + -1.5 = -6.946221995

Simplifying
x + -1.5 = -6.946221995

Reorder the terms:
-1.5 + x = -6.946221995

Solving
-1.5 + x = -6.946221995

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -6.946221995 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -6.946221995 + 1.5
x = -6.946221995 + 1.5

Combine like terms: -6.946221995 + 1.5 = -5.446221995
x = -5.446221995

Simplifying
x = -5.446221995

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.446221995, -5.446221995}
Solution
x = {8.446221995, -5.446221995}