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To simplify the given polynomial expression, let's go through it step-by-step. The expression is:

\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)
\]

### Step 1: Simplify each part of the expression.

1. **First Group:** \(3x^2 - x - 7\)
2. **Second Group (with subtraction):** \(-(5x^2 - 4x - 2)\)
   - Distribute the negative sign: \(-5x^2 + 4x + 2\)
3. **Third Group:** Expand \((x + 3)(x + 2)\) using the distributive property:
   \[
   (x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
   \]

### Step 2: Combine all terms.

Now we have:
\[
3x^2 - x - 7 - 5x^2 + 4x + 2 + x^2 + 5x + 6
\]

Combine like terms:
- \(3x^2 - 5x^2 + x^2 = -x^2\)
- \(-x + 4x + 5x = 8x\)
- \(-7 + 2 + 6 = 1\)

### Simplified Expression

The simplified form of the polynomial is:
\[
-x^2 + 8x + 1
\]

### Degree and Type of Polynomial

- **Type of Polynomial:** It’s a **trinomial** because it has three terms.
- **Degree:** The highest power of \(x\) is 2, so the degree is **2**.

### Answer

The polynomial simplifies to an expression that is a **trinomial** with a degree of **2**.

Would you like a deeper explanation or have any questions?

Here are five related questions you may find helpful:
1. How do you multiply binomials using the distributive property?
2. What are like terms in polynomial expressions, and how are they combined?
3. How do you determine the degree of a polynomial?
4. Why is distributing the negative sign important in polynomial simplification?
5. What classifies a polynomial as a monomial, binomial, or trinomial?

**Tip:** When simplifying polynomials, always handle any negative signs carefully to avoid sign errors.

Simplify the polynomial expression: (3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2). Determine the type of the simplified polynomial and its degree.

Learn how to simplify the polynomial expression (3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2) with a detailed, step-by-step solution. This example covers polynomial simplification using the distributive property and combining like terms, suitable for students in Grades 8-10.

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