Let's simplify the given polynomial expression step-by-step:
Given polynomial expression:
[tex]\[
\left(3 x^2 - x - 7\right) - \left(5 x^2 - 4 x - 2\right) + (x + 3)(x + 2)
\][/tex]
First, distribute the subtraction across the second polynomial:
[tex]\[
(3 x^2 - x - 7) - 5 x^2 + 4 x + 2
\][/tex]
Next, expand the product [tex]\((x + 3)(x + 2)\)[/tex]:
[tex]\[
(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
\][/tex]
Now, combine all parts of the expression:
[tex]\[
3 x^2 - x - 7 - 5 x^2 + 4 x + 2 + x^2 + 5 x + 6
\][/tex]
Combine like terms:
[tex]\[
(3 x^2 - 5 x^2 + x^2) + (-x + 4 x + 5 x) + (-7 + 2 + 6)
\][/tex]
This simplifies to:
[tex]\[
-x^2 + 8x + 1
\][/tex]
The simplified polynomial expression is:
[tex]\[
-x^2 + 8x + 1
\][/tex]
Now, let's determine the type and degree of this polynomial.
The highest power of [tex]\( x \)[/tex] in the simplified expression is [tex]\( x^2 \)[/tex], which means the degree of the polynomial is 2. A polynomial with a degree of 2 is called a quadratic polynomial.
Therefore, the polynomial simplifies to an expression that is a:
[tex]\[
\text{quadratic}
\][/tex]
with a degree of:
[tex]\[
2
\][/tex]
So, the complete answer is:
The polynomial simplifies to an expression that is a [tex]\(\text{quadratic}\)[/tex] with a degree of [tex]\(2\)[/tex].
