10x^4+30x^3+10x^2=4x^2-x^3

Solution for 10x^4+30x^3+10x^2=4x^2-x^3 equation:

Simplifying
10x4 + 30x3 + 10x2 = 4x2 + -1x3

Reorder the terms:
10x2 + 30x3 + 10x4 = 4x2 + -1x3

Solving
10x2 + 30x3 + 10x4 = 4x2 + -1x3

Solving for variable 'x'.

Reorder the terms:
10x2 + -4x2 + 30x3 + x3 + 10x4 = 4x2 + -1x3 + -4x2 + x3

Combine like terms: 10x2 + -4x2 = 6x2
6x2 + 30x3 + x3 + 10x4 = 4x2 + -1x3 + -4x2 + x3

Combine like terms: 30x3 + x3 = 31x3
6x2 + 31x3 + 10x4 = 4x2 + -1x3 + -4x2 + x3

Reorder the terms:
6x2 + 31x3 + 10x4 = 4x2 + -4x2 + -1x3 + x3

Combine like terms: 4x2 + -4x2 = 0
6x2 + 31x3 + 10x4 = 0 + -1x3 + x3
6x2 + 31x3 + 10x4 = -1x3 + x3

Combine like terms: -1x3 + x3 = 0
6x2 + 31x3 + 10x4 = 0

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(6 + 31x + 10x2) = 0

Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve:

Simplifying
x2 = 0

Solving
x2 = 0

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

Simplifying
x2 = 0

Take the square root of each side:
x = {0}
Subproblem 2
Set the factor '(6 + 31x + 10x2)' equal to zero and attempt to solve:

Simplifying
6 + 31x + 10x2 = 0

Solving
6 + 31x + 10x2 = 0

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

Divide each side by '10'.
0.6 + 3.1x + x2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 3.1x + -0.6 + x2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 3.1x + x2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 3.1x + x2 = 0 + -0.6
3.1x + x2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
3.1x + x2 = -0.6

The x term is 3.1x.  Take half its coefficient (1.55).
Square it (2.4025) and add it to both sides.

Add '2.4025' to each side of the equation.
3.1x + 2.4025 + x2 = -0.6 + 2.4025

Reorder the terms:
2.4025 + 3.1x + x2 = -0.6 + 2.4025

Combine like terms: -0.6 + 2.4025 = 1.8025
2.4025 + 3.1x + x2 = 1.8025

Factor a perfect square on the left side:
(x + 1.55)(x + 1.55) = 1.8025

Calculate the square root of the right side: 1.342572158

Break this problem into two subproblems by setting 
(x + 1.55) equal to 1.342572158 and -1.342572158.

Subproblem 1
x + 1.55 = 1.342572158

Simplifying
x + 1.55 = 1.342572158

Reorder the terms:
1.55 + x = 1.342572158

Solving
1.55 + x = 1.342572158

Solving for variable 'x'.

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

Add '-1.55' to each side of the equation.
1.55 + -1.55 + x = 1.342572158 + -1.55

Combine like terms: 1.55 + -1.55 = 0.00
0.00 + x = 1.342572158 + -1.55
x = 1.342572158 + -1.55

Combine like terms: 1.342572158 + -1.55 = -0.207427842
x = -0.207427842

Simplifying
x = -0.207427842

Subproblem 2
x + 1.55 = -1.342572158

Simplifying
x + 1.55 = -1.342572158

Reorder the terms:
1.55 + x = -1.342572158

Solving
1.55 + x = -1.342572158

Solving for variable 'x'.

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

Add '-1.55' to each side of the equation.
1.55 + -1.55 + x = -1.342572158 + -1.55

Combine like terms: 1.55 + -1.55 = 0.00
0.00 + x = -1.342572158 + -1.55
x = -1.342572158 + -1.55

Combine like terms: -1.342572158 + -1.55 = -2.892572158
x = -2.892572158

Simplifying
x = -2.892572158

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.207427842, -2.892572158}
Solution
x = {0, -0.207427842, -2.892572158}