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Simplify the following polynomial expression:
[tex]\[
(3x^2 - x - 7) - (5x^2 - 4x - 2) + (x + 3)(x + 2)
\][/tex]

The polynomial simplifies to an expression that is a [tex]$\square$[/tex] [tex]$\square$[/tex] with a degree of [tex]$\square$[/tex].



Answer :

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].