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.