https://file.aiquickdraw.com/mathbot/file/1730138003090k7pr1tiz.jpg

Let's simplify the expression step by step:

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

### Step 1: Expand \((x + 3)(x + 2)\)
\[
(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\]

Now substitute this back into the expression:
\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x^2 + 5x + 6)
\]

### Step 2: Distribute the Negative Sign
Rewrite the expression by distributing the negative sign in front of \((5x^2 - 4x - 2)\):
\[
= 3x^2 - x - 7 - 5x^2 + 4x + 2 + x^2 + 5x + 6
\]

### Step 3: Combine Like Terms
Now, group the terms by their powers of \(x\):
- For \(x^2\): \(3x^2 - 5x^2 + x^2 = -x^2\)
- For \(x\): \(-x + 4x + 5x = 8x\)
- For the constants: \(-7 + 2 + 6 = 1\)

So, the simplified expression is:
\[
-x^2 + 8x + 1
\]

### Step 4: Determine the Type and Degree
The resulting expression is a **polynomial** with the highest power of \(x\) being 2. This makes it a **quadratic polynomial** with a degree of **2**.

**Answer:**
The polynomial simplifies to an expression that is a **quadratic** with a degree of **2**.

Would you like more details or have any questions?

Here are some related questions:
1. What are the steps to expand binomials?
2. How can we identify the degree of a polynomial?
3. What is the significance of the degree of a polynomial?
4. How does distributing negative signs affect polynomial simplification?
5. What are like terms, and why is it important to combine them in simplification?

**Tip:** When simplifying expressions, carefully distribute any negative signs to avoid sign errors.

Simplify the following polynomial expression:

(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)

The polynomial simplifies to an expression that is a ___ with a degree of ___.

Learn how to simplify a polynomial expression and identify its type and degree. This problem includes polynomial expansion, combining like terms, and recognizing quadratic expressions. Suitable for students in Grades 8-10.

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