To simplify the polynomial expression [tex]\((3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)\)[/tex], follow these steps:
1. Distribute the negative sign through the second polynomial:
[tex]\[
(3x^2 - x - 7) - 5x^2 + 4x + 2 + (x + 3)(x + 2)
\][/tex]
2. Expand the product [tex]\((x + 3)(x + 2)\)[/tex]:
[tex]\[
(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\][/tex]
3. Combine like terms of all the polynomials:
[tex]\[
3x^2 - x - 7 - 5x^2 + 4x + 2 + x^2 + 5x + 6
\][/tex]
4. Collect all [tex]\(x^2\)[/tex] terms:
[tex]\[
3x^2 - 5x^2 + x^2 = -x^2
\][/tex]
5. Collect all [tex]\(x\)[/tex] terms:
[tex]\[
-x + 4x + 5x = 8x
\][/tex]
6. Collect all constant terms:
[tex]\[
-7 + 2 + 6 = 1
\][/tex]
7. Combine the simplified terms to get the final polynomial:
[tex]\[
-x^2 + 8x + 1
\][/tex]
The polynomial simplifies to an expression that is a
[tex]\[
- \quad x^2 + 8x + 1
\][/tex]
which is a quadratic polynomial with a degree of [tex]\(2\)[/tex].
Thus:
- The polynomial simplifies to an expression that is a [tex]\(\boxed{\text{quadratic polynomial}}\)[/tex] with a degree of [tex]\(\boxed{2}\)[/tex].
