x(2x+1)(x+5)-(2x^2+1)(x+4.5)=3.5

Solution for x(2x+1)(x+5)-(2x^2+1)(x+4.5)=3.5 equation:

Simplifying
x(2x + 1)(x + 5) + -1(2x2 + 1)(x + 4.5) = 3.5

Reorder the terms:
x(1 + 2x)(x + 5) + -1(2x2 + 1)(x + 4.5) = 3.5

Reorder the terms:
x(1 + 2x)(5 + x) + -1(2x2 + 1)(x + 4.5) = 3.5

Multiply (1 + 2x) * (5 + x)
x(1(5 + x) + 2x * (5 + x)) + -1(2x2 + 1)(x + 4.5) = 3.5
x((5 * 1 + x * 1) + 2x * (5 + x)) + -1(2x2 + 1)(x + 4.5) = 3.5
x((5 + 1x) + 2x * (5 + x)) + -1(2x2 + 1)(x + 4.5) = 3.5
x(5 + 1x + (5 * 2x + x * 2x)) + -1(2x2 + 1)(x + 4.5) = 3.5
x(5 + 1x + (10x + 2x2)) + -1(2x2 + 1)(x + 4.5) = 3.5

Combine like terms: 1x + 10x = 11x
x(5 + 11x + 2x2) + -1(2x2 + 1)(x + 4.5) = 3.5
(5 * x + 11x * x + 2x2 * x) + -1(2x2 + 1)(x + 4.5) = 3.5
(5x + 11x2 + 2x3) + -1(2x2 + 1)(x + 4.5) = 3.5

Reorder the terms:
5x + 11x2 + 2x3 + -1(1 + 2x2)(x + 4.5) = 3.5

Reorder the terms:
5x + 11x2 + 2x3 + -1(1 + 2x2)(4.5 + x) = 3.5

Multiply (1 + 2x2) * (4.5 + x)
5x + 11x2 + 2x3 + -1(1(4.5 + x) + 2x2 * (4.5 + x)) = 3.5
5x + 11x2 + 2x3 + -1((4.5 * 1 + x * 1) + 2x2 * (4.5 + x)) = 3.5
5x + 11x2 + 2x3 + -1((4.5 + 1x) + 2x2 * (4.5 + x)) = 3.5
5x + 11x2 + 2x3 + -1(4.5 + 1x + (4.5 * 2x2 + x * 2x2)) = 3.5
5x + 11x2 + 2x3 + -1(4.5 + 1x + (9x2 + 2x3)) = 3.5
5x + 11x2 + 2x3 + -1(4.5 + 1x + 9x2 + 2x3) = 3.5
5x + 11x2 + 2x3 + (4.5 * -1 + 1x * -1 + 9x2 * -1 + 2x3 * -1) = 3.5
5x + 11x2 + 2x3 + (-4.5 + -1x + -9x2 + -2x3) = 3.5

Reorder the terms:
-4.5 + 5x + -1x + 11x2 + -9x2 + 2x3 + -2x3 = 3.5

Combine like terms: 5x + -1x = 4x
-4.5 + 4x + 11x2 + -9x2 + 2x3 + -2x3 = 3.5

Combine like terms: 11x2 + -9x2 = 2x2
-4.5 + 4x + 2x2 + 2x3 + -2x3 = 3.5

Combine like terms: 2x3 + -2x3 = 0
-4.5 + 4x + 2x2 + 0 = 3.5
-4.5 + 4x + 2x2 = 3.5

Solving
-4.5 + 4x + 2x2 = 3.5

Solving for variable 'x'.

Reorder the terms:
-4.5 + -3.5 + 4x + 2x2 = 3.5 + -3.5

Combine like terms: -4.5 + -3.5 = -8
-8 + 4x + 2x2 = 3.5 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
-8 + 4x + 2x2 = 0.0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + 2x + x2) = 0.0

Ignore the factor 2.
Subproblem 1
Set the factor '(-4 + 2x + x2)' equal to zero and attempt to solve:

Simplifying
-4 + 2x + x2 = 0

Solving
-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}
Solution
x = {1.236067978, -3.236067978}