Let's simplify the given expression and classify the resulting polynomial step by step:
1. Given Expression:
[tex]\[
3x(x-3) + (2x + 6)(-x - 3)
\][/tex]
2. Expanding Each Term:
- For the first part, [tex]\(3x(x-3)\)[/tex]:
[tex]\[
3x \cdot x - 3x \cdot 3 = 3x^2 - 9x
\][/tex]
- For the second part, [tex]\((2x + 6)(-x - 3)\)[/tex]:
[tex]\[
(2x)(-x) + (2x)(-3) + (6)(-x) + (6)(-3) = -2x^2 - 6x - 6x - 18 = -2x^2 - 12x - 18
\][/tex]
3. Combining the Expanded Terms:
Add the results from the first and second parts together:
[tex]\[
3x^2 - 9x + (-2x^2 - 12x - 18)
\][/tex]
Simplify by combining like terms:
[tex]\[
(3x^2 - 2x^2) + (-9x - 12x) + (-18) = x^2 - 21x - 18
\][/tex]
4. Resulting Polynomial:
The simplified expression is:
[tex]\[
x^2 - 21x - 18
\][/tex]
5. Classifying the Polynomial:
- The degree of the polynomial [tex]\(x^2 - 21x - 18\)[/tex] is 2 (since the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]).
- It contains three terms: [tex]\(x^2\)[/tex], [tex]\(-21x\)[/tex], and [tex]\(-18\)[/tex].
Since it is a second-degree polynomial with three terms, the correct classification is:
Quadratic Trinomial
Thus, the correct answer is:
- Quadratic trinomial.
