Sure, let's simplify the given polynomial expression step-by-step.
Given expression:
[tex]\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)
\][/tex]
First, let's distribute and simplify each part separately.
1. [tex]\((3x^2 - x - 7)\)[/tex]
2. Subtract the second polynomial: [tex]\((5x^2 - 4x - 2)\)[/tex]
[tex]\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) = 3x^2 - x - 7 - 5x^2 + 4x + 2
\][/tex]
Combine like terms:
[tex]\[
(3x^2 - 5x^2) + (-x + 4x) + (-7 + 2) = -2x^2 + 3x - 5
\][/tex]
3. Expand the product [tex]\((x + 3)(x + 2)\)[/tex]:
[tex]\[
(x + 3)(x + 2) = x^2 + 2x + 3x + 6
\][/tex]
Combine like terms:
[tex]\[
x^2 + 5x + 6
\][/tex]
Now add this expanded product to the simplified form from earlier:
[tex]\[
(-2x^2 + 3x - 5) + (x^2 + 5x + 6)
\][/tex]
Combine like terms:
[tex]\[
(-2x^2 + x^2) + (3x + 5x) + (-5 + 6) = -x^2 + 8x + 1
\][/tex]
So, the polynomial simplifies to:
[tex]\[
-x^2 + 8x + 1
\][/tex]
The degree of this polynomial is the highest power of [tex]\(x\)[/tex] in the expression, which is 2.
Hence, the polynomial simplifies to an expression that is a quadratic polynomial with a degree of 2.
