To simplify the polynomial expression [tex]\((3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)\)[/tex], let's go through it step by step.
1. First, expand [tex]\((x + 3)(x + 2)\)[/tex]:
[tex]\[
(x + 3)(x + 2) = x \cdot x + x \cdot 2 + 3 \cdot x + 3 \cdot 2 = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\][/tex]
2. Next, distribute the negative sign in [tex]\(-(5x^2 - 4x - 2)\)[/tex]:
[tex]\[
- (5x^2 - 4x - 2) = -5x^2 + 4x + 2
\][/tex]
3. Now, combine all parts of the expression:
[tex]\[
(3x^2 - x - 7) + (-5x^2 + 4x + 2) + (x^2 + 5x + 6)
\][/tex]
4. Combine like terms for each degree of [tex]\(x\)[/tex]:
- For [tex]\(x^2\)[/tex]:
[tex]\[
3x^2 - 5x^2 + x^2 = -x^2
\][/tex]
- For [tex]\(x\)[/tex]:
[tex]\[
-x + 4x + 5x = 8x
\][/tex]
- For the constant term:
[tex]\[
-7 + 2 + 6 = 1
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
-x^2 + 8x + 1
\][/tex]
The simplified polynomial is a trinomial with a degree of 2.
Hence, the polynomial simplifies to an expression that is a trinomial with a degree of 2.
