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

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

Simplifying
0 = x4 + 2x3 + x2

Reorder the terms:
0 = x2 + 2x3 + x4

Solving
0 = x2 + 2x3 + x4

Solving for variable 'x'.
Remove the zero:
-1x2 + -2x3 + -1x4 = x2 + 2x3 + x4 + -1x2 + -2x3 + -1x4

Reorder the terms:
-1x2 + -2x3 + -1x4 = x2 + -1x2 + 2x3 + -2x3 + x4 + -1x4

Combine like terms: x2 + -1x2 = 0
-1x2 + -2x3 + -1x4 = 0 + 2x3 + -2x3 + x4 + -1x4
-1x2 + -2x3 + -1x4 = 2x3 + -2x3 + x4 + -1x4

Combine like terms: 2x3 + -2x3 = 0
-1x2 + -2x3 + -1x4 = 0 + x4 + -1x4
-1x2 + -2x3 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-1x2 + -2x3 + -1x4 = 0

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

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

Ignore the factor -1.
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}