(x)*(x+1)=31-(x-2)

Solution for (x)*(x+1)=31-(x-2) equation:

Simplifying
(x)(x + 1) = 31 + -1(x + -2)

Reorder the terms:
x(1 + x) = 31 + -1(x + -2)
(1 * x + x * x) = 31 + -1(x + -2)
(1x + x2) = 31 + -1(x + -2)

Reorder the terms:
1x + x2 = 31 + -1(-2 + x)
1x + x2 = 31 + (-2 * -1 + x * -1)
1x + x2 = 31 + (2 + -1x)

Combine like terms: 31 + 2 = 33
1x + x2 = 33 + -1x

Solving
1x + x2 = 33 + -1x

Solving for variable 'x'.

Reorder the terms:
-33 + 1x + x + x2 = 33 + -1x + -33 + x

Combine like terms: 1x + x = 2x
-33 + 2x + x2 = 33 + -1x + -33 + x

Reorder the terms:
-33 + 2x + x2 = 33 + -33 + -1x + x

Combine like terms: 33 + -33 = 0
-33 + 2x + x2 = 0 + -1x + x
-33 + 2x + x2 = -1x + x

Combine like terms: -1x + x = 0
-33 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '33' to each side of the equation.
-33 + 2x + 33 + x2 = 0 + 33

Reorder the terms:
-33 + 33 + 2x + x2 = 0 + 33

Combine like terms: -33 + 33 = 0
0 + 2x + x2 = 0 + 33
2x + x2 = 0 + 33

Combine like terms: 0 + 33 = 33
2x + x2 = 33

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 33 + 1

Reorder the terms:
1 + 2x + x2 = 33 + 1

Combine like terms: 33 + 1 = 34
1 + 2x + x2 = 34

Factor a perfect square on the left side:
(x + 1)(x + 1) = 34

Calculate the square root of the right side: 5.830951895

Break this problem into two subproblems by setting 
(x + 1) equal to 5.830951895 and -5.830951895.

Subproblem 1
x + 1 = 5.830951895

Simplifying
x + 1 = 5.830951895

Reorder the terms:
1 + x = 5.830951895

Solving
1 + x = 5.830951895

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 5.830951895 + -1

Combine like terms: 1 + -1 = 0
0 + x = 5.830951895 + -1
x = 5.830951895 + -1

Combine like terms: 5.830951895 + -1 = 4.830951895
x = 4.830951895

Simplifying
x = 4.830951895

Subproblem 2
x + 1 = -5.830951895

Simplifying
x + 1 = -5.830951895

Reorder the terms:
1 + x = -5.830951895

Solving
1 + x = -5.830951895

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -5.830951895 + -1

Combine like terms: 1 + -1 = 0
0 + x = -5.830951895 + -1
x = -5.830951895 + -1

Combine like terms: -5.830951895 + -1 = -6.830951895
x = -6.830951895

Simplifying
x = -6.830951895

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.830951895, -6.830951895}