0=5x^4+10x^3+5x^2

Solution for 0=5x^4+10x^3+5x^2 equation:

Simplifying
0 = 5x4 + 10x3 + 5x2

Reorder the terms:
0 = 5x2 + 10x3 + 5x4

Solving
0 = 5x2 + 10x3 + 5x4

Solving for variable 'x'.
Remove the zero:
-5x2 + -10x3 + -5x4 = 5x2 + 10x3 + 5x4 + -5x2 + -10x3 + -5x4

Reorder the terms:
-5x2 + -10x3 + -5x4 = 5x2 + -5x2 + 10x3 + -10x3 + 5x4 + -5x4

Combine like terms: 5x2 + -5x2 = 0
-5x2 + -10x3 + -5x4 = 0 + 10x3 + -10x3 + 5x4 + -5x4
-5x2 + -10x3 + -5x4 = 10x3 + -10x3 + 5x4 + -5x4

Combine like terms: 10x3 + -10x3 = 0
-5x2 + -10x3 + -5x4 = 0 + 5x4 + -5x4
-5x2 + -10x3 + -5x4 = 5x4 + -5x4

Combine like terms: 5x4 + -5x4 = 0
-5x2 + -10x3 + -5x4 = 0

Factor out the Greatest Common Factor (GCF), '-5x2'.
-5x2(1 + 2x + x2) = 0

Factor a trinomial.
-5x2((1 + x)(1 + x)) = 0

Ignore the factor -5.
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 '(1 + x)' equal to zero and attempt to solve:

Simplifying
1 + x = 0

Solving
1 + x = 0

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 = 0 + -1

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

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

Simplifying
x = -1
Subproblem 3
Set the factor '(1 + x)' equal to zero and attempt to solve:

Simplifying
1 + x = 0

Solving
1 + x = 0

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 = 0 + -1

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

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

Simplifying
x = -1
Solution
x = {0, -1, -1}