https://file.aiquickdraw.com/mathbot/file/1728415600553p5tv61ob.jpg

The inequality provided in the image is:

\[
x^3 + 2x^2 + x + 2 > 0
\]

Let's solve this inequality step by step.

### 1. Factor the cubic expression if possible:
We attempt factoring \( x^3 + 2x^2 + x + 2 \) using methods such as synthetic division or trial values for potential roots.

#### Step 1: Trial and Error with Rational Roots
Let's test small integer values for \(x\). Substituting \(x = -1\):

\[
(-1)^3 + 2(-1)^2 + (-1) + 2 = -1 + 2 - 1 + 2 = 2 \neq 0
\]

\(x = -1\) is not a root. Now, test \(x = -2\):

\[
(-2)^3 + 2(-2)^2 + (-2) + 2 = -8 + 8 - 2 + 2 = 0
\]

\(x = -2\) is a root of the cubic equation.

#### Step 2: Synthetic Division
Now that we know \(x = -2\) is a root, we divide \(x^3 + 2x^2 + x + 2\) by \(x + 2\):

\[
x^3 + 2x^2 + x + 2 \div (x + 2)
\]

The quotient is \(x^2 + 0x + 1\). Therefore, the factorization is:

\[
x^3 + 2x^2 + x + 2 = (x + 2)(x^2 + 1)
\]

### 2. Solve the Inequality
Now we have:

\[
(x + 2)(x^2 + 1) > 0
\]

#### Analyze Each Factor:
- \(x + 2 > 0\) when \(x > -2\).
- \(x^2 + 1 > 0\) for all real \(x\) since \(x^2 + 1\) is always positive.

Thus, the inequality holds when \(x + 2 > 0\), i.e., \(x > -2\).

### 3. Final Answer:
The solution to the inequality \(x^3 + 2x^2 + x + 2 > 0\) is:

\[
x > -2
\]

Would you like a detailed explanation of the synthetic division or have any questions?

Here are 5 related questions for practice:
1. What are the steps to factor a cubic polynomial?
2. How do you perform synthetic division with polynomials?
3. What is the significance of \(x^2 + 1 > 0\) in this problem?
4. How would the solution change if the inequality was \( \leq \)?
5. Can a cubic polynomial have more than one real root?

**Tip:** When solving polynomial inequalities, always check if the expression can be factored, and use synthetic division to simplify the process.

Solve the inequality: x^3 + 2x^2 + x + 2 > 0

Learn how to solve the polynomial inequality x^3 + 2x^2 + x + 2 > 0 with a step-by-step explanation. This guide covers algebraic concepts such as polynomial factorization and synthetic division, suitable for students in Grades 10-12.

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