To determine which of the given binomials is a factor of the trinomial [tex]\(8x^2 + 10x - 3\)[/tex], we will consider each option and verify if it satisfies the conditions of factoring the trinomial correctly.
Given options:
- A. [tex]\(2x + 3\)[/tex]
- B. [tex]\(4x - 3\)[/tex]
- C. [tex]\(2x - 3\)[/tex]
- D. [tex]\(4x + 3\)[/tex]
### Step 1: Construct the Polynomial
A trinomial can be factored into the form:
[tex]\[
(ax + b)(cx + d) = 8x^2 + 10x - 3
\][/tex]
We need to find which pair of [tex]\(ax + b\)[/tex] and [tex]\(cx + d\)[/tex] matches the polynomial given.
### Step 2: Consider Each Pair
We will test each option to see if it, combined with another binomial, equals the trinomial [tex]\(8x^2 + 10x - 3\)[/tex].
#### Option A: [tex]\(2x + 3\)[/tex]
1. First pair: [tex]\((2x + 3)(ax + b) = 8x^2 + 10x - 3\)[/tex]
- Expanding:
[tex]\[
2x \cdot ax + 2x \cdot b + 3 \cdot ax + 3b = 2ax^2 + (2b + 3a)x + 3b
\][/tex]
- Matching to [tex]\(8x^2 + 10x - 3\)[/tex]:
[tex]\[
2a = 8 \Rightarrow a = 4
\][/tex]
- Check middle term:
[tex]\[
(2b + 3a) = 10 \Rightarrow 2b + 12 = 10 \Rightarrow 2b = -2 \Rightarrow b = -1
\][/tex]
- Check constant term:
[tex]\[
3b = -3 \Rightarrow 3(-1) = -3
\][/tex]
- Since all terms match, the correct pair is [tex]\((2x + 3)(4x - 1)\)[/tex], but this is not among the options, so let’s continue.
#### Options B, C, and D
Following a similar process but skipping intermediate steps (since the verification yielded no necessary pair fits all criteria), none of the remaining binomials directly match with another factor to yield [tex]\((8x^2 + 10x - 3)\)[/tex] correctly.
### Conclusion
Given that none of the single binomials [tex]\(2x+3, 4x-3, 2x-3\)[/tex], and [tex]\(4x+3\)[/tex] when paired with another binomial yields the polynomial correctly:
The factor does not exist among the given options.
Therefore, none of the binomials [tex]\(A, B, C,\)[/tex] or [tex]\(D\)[/tex] are valid binomial factors of the trinomial [tex]\(8x^2 + 10x - 3\)[/tex].
