x+x^2+x+1=5

Solution for x+x^2+x+1=5 equation:

Simplifying
x + x2 + x + 1 = 5

Reorder the terms:
1 + x + x + x2 = 5

Combine like terms: x + x = 2x
1 + 2x + x2 = 5

Solving
1 + 2x + x2 = 5

Solving for variable 'x'.

Reorder the terms:
1 + -5 + 2x + x2 = 5 + -5

Combine like terms: 1 + -5 = -4
-4 + 2x + x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-4 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 2x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2x + x2 = 0 + 4
2x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
2x + x2 = 4

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 = 4 + 1

Reorder the terms:
1 + 2x + x2 = 4 + 1

Combine like terms: 4 + 1 = 5
1 + 2x + x2 = 5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + 1) equal to 2.236067978 and -2.236067978.

Subproblem 1
x + 1 = 2.236067978

Simplifying
x + 1 = 2.236067978

Reorder the terms:
1 + x = 2.236067978

Solving
1 + x = 2.236067978

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 = 2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + x = 2.236067978 + -1
x = 2.236067978 + -1

Combine like terms: 2.236067978 + -1 = 1.236067978
x = 1.236067978

Simplifying
x = 1.236067978

Subproblem 2
x + 1 = -2.236067978

Simplifying
x + 1 = -2.236067978

Reorder the terms:
1 + x = -2.236067978

Solving
1 + x = -2.236067978

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 = -2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + x = -2.236067978 + -1
x = -2.236067978 + -1

Combine like terms: -2.236067978 + -1 = -3.236067978
x = -3.236067978

Simplifying
x = -3.236067978

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.236067978, -3.236067978}