To simplify the expression [tex]\((8x^2 + 3x) - (12x^2 - 1)\)[/tex] and classify the resulting polynomial, we can follow these specific steps:
1. Rewrite the Expression:
[tex]\[
(8x^2 + 3x) - (12x^2 - 1)
\][/tex]
2. Distribute the Negative Sign:
We need to apply the negative sign to each term inside the parentheses that follows it:
[tex]\[
8x^2 + 3x - 12x^2 + 1
\][/tex]
3. Combine Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 - 12x^2\)[/tex]
- Combine the constant terms and remaining linear terms: [tex]\(3x + 1\)[/tex]
So we get:
[tex]\[
(8x^2 - 12x^2) + 3x + 1
\][/tex]
Simplify the [tex]\(x^2\)[/tex] terms:
[tex]\[
-4x^2 + 3x + 1
\][/tex]
4. Classification of the Resulting Polynomial:
- The expression [tex]\(-4x^2 + 3x + 1\)[/tex] includes three terms: one quadratic term [tex]\(-4x^2\)[/tex], one linear term [tex]\(3x\)[/tex], and one constant term [tex]\(1\)[/tex].
- Since it contains a term with [tex]\(x^2\)[/tex] (a quadratic term), a term with [tex]\(x\)[/tex] (a linear term), and a constant term, it has exactly three terms.
This type of polynomial is known as a quadratic trinomial.
Therefore, the correct classification is:
[tex]\[
\boxed{A \text{. quadratic trinomial}}
\][/tex]
