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The inequality provided is:

\[
3x^2 + 2x < x^4
\]

To solve this polynomial inequality, follow these steps:

### Step 1: Rearrange the inequality
Move all terms to one side of the inequality so that the other side is 0:

\[
3x^2 + 2x - x^4 < 0
\]

### Step 2: Factor the polynomial
Rearrange the terms in descending powers of \( x \):

\[
-x^4 + 3x^2 + 2x < 0
\]

Factor out a negative sign:

\[
-(x^4 - 3x^2 - 2x) < 0
\]

Now, factor the polynomial inside the parentheses. Begin by factoring out \( x \):

\[
-x(x^3 - 3x - 2) < 0
\]

Next, factor the cubic polynomial \( x^3 - 3x - 2 \). Use trial and error or synthetic division to find its factors. Checking with \( x = -2 \) gives:

\[
(-2)^3 - 3(-2) - 2 = -8 + 6 - 2 = -4
\]

Checking with \( x = 1 \) gives:

\[
(1)^3 - 3(1) - 2 = 1 - 3 - 2 = -4
\]

Further factoring gives:

\[
x(x + 1)(x - 2) = 0
\]


Solve the polynomial inequality 3x^2 + 2x < x^4 and state the answer using interval notation.

Learn how to solve the polynomial inequality 3x^2 + 2x < x^4. This problem covers key algebraic concepts such as factoring polynomials, rearranging inequalities, and using the zero product property to find critical points. Suitable for students in Grades 10-12.

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