The product of a binomial and a trinomial is *x* 3 + 3 *x* 2 − *x* + 2 *x* 2 + 6 *x* − 2. Which expression is equivalent to this product after it has been fully simplified?

*x*3 + 5*x*2  + 5*x* – 2

*x*3 + 2*x*2 + 8*x* – 2

*x*3 + 11*x*2 – 2

*x*3 + 10*x*2 – 2

To solve this problem, we need to first simplify the expression given, and then compare the simplified result with the provided options.

The expression given is:
\[
x^3 + 3x^2 - x + 2x^2 + 6x - 2
\]

### Step 1: Combine like terms
- \(x^3\) term: There's only one \(x^3\) term, so it stays as \(x^3\).
- \(x^2\) terms: \(3x^2 + 2x^2 = 5x^2\)
- \(x\) terms: \(-x + 6x = 5x\)
- Constant term: \(-2\)

### Step 2: Write the simplified expression
\[
x^3 + 5x^2 + 5x - 2
\]

### Step 3: Compare with the provided options

The simplified expression matches the first option:
\[
\boxed{x^3 + 5x^2 + 5x - 2}
\]

This is the correct equivalent expression.

Do you want further details or have any questions?

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Here are 5 related questions you might find useful:
1. How do you expand the product of a binomial and a trinomial?
2. What are like terms, and how do you combine them?
3. How do you factor a cubic polynomial like \(x^3 + 5x^2 + 5x - 2\)?
4. What is the significance of the constant term in a polynomial expression?
5. How can polynomial expressions be used in real-world applications?

**Tip:** When simplifying expressions, always look for like terms to combine. This often makes the expression easier to understand and solve.

Learn how to simplify the product of a binomial and a trinomial to its equivalent expression. This problem involves combining like terms and understanding polynomial simplification. Suitable for Grades 9-12 students.

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