Sure, let's break down the given polynomial expression step by step.
We're given the expression:
[tex]\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)
\][/tex]
1. Distribute and Simplify the Subtracted Terms:
- Start by expanding the subtraction in the first part.
[tex]\[
(3x^2 - x - 7) - (5x^2 - 4x - 2)
\][/tex]
- Distribute the negative sign through the second polynomial.
[tex]\[
3x^2 - x - 7 - 5x^2 + 4x + 2
\][/tex]
- Combine like terms.
[tex]\[
3x^2 - 5x^2 + (-x + 4x) + (-7 + 2)
\][/tex]
[tex]\[
-2x^2 + 3x - 5
\][/tex]
2. Expand the Multiplication:
- Now, expand [tex]\((x + 3)(x + 2)\)[/tex].
[tex]\[
(x + 3)(x + 2) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
\][/tex]
3. Combine Everything:
- Add the results from the previous steps.
[tex]\[
(-2x^2 + 3x - 5) + (x^2 + 5x + 6)
\][/tex]
- Combine the like terms.
[tex]\[
-2x^2 + x^2 + 3x + 5x - 5 + 6
\][/tex]
[tex]\[
-x^2 + 8x + 1
\][/tex]
4. Identify the Degree:
- The resulting polynomial is [tex]\(-x^2 + 8x + 1\)[/tex].
- The degree of the polynomial is the highest power of [tex]\(x\)[/tex], which is 2.
So, the polynomial simplifies to an expression that is a [tex]\(-x^2 + 8x + 1\)[/tex] with a degree of 2.
