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.