To determine which binomial is a factor of the given trinomial [tex]\( 8x^2 + 10x - 3 \)[/tex], we need to examine each binomial option and check if they divide the trinomial without leaving a remainder.
### Trinomial:
[tex]\[ 8x^2 + 10x - 3 \][/tex]
### Binomial options:
A. [tex]\( 2x + 3 \)[/tex]
B. [tex]\( 4x - 3 \)[/tex]
C. [tex]\( 2x - 3 \)[/tex]
D. [tex]\( 4x + 3 \)[/tex]
### Step-by-Step Solution:
1. Check if [tex]\(2x + 3\)[/tex] is a factor of [tex]\(8x^2 + 10x - 3\)[/tex]:
To test if [tex]\(2x + 3\)[/tex] is a factor, perform polynomial division or determine if the product of the factors equals the trinomial.
2. Determine factorization:
To factorize [tex]\(8x^2 + 10x - 3\)[/tex], we need to see if it's the product of two binomials. Let's try:
Suppose [tex]\( 8x^2 + 10x - 3 \)[/tex] can be expressed as [tex]\( (ax + b)(cx + d) \)[/tex].
By observing [tex]\( 2x + 3 \)[/tex], let's check if:
[tex]\[
(2x + 3)(something) = 8x^2 + 10x - 3
\][/tex]
3. Testing [tex]\( 2x + 3 \)[/tex]:
The correct factorization involves risk calculations but let’s proceed with how the factorization would appear:
[tex]\[
(2x + 3)(...) = 8x^2 + 10x - 3
\][/tex]
Simplifying:
[tex]\[
8x^2 + kx - 9
\][/tex]
Matching coefficients, we can derive specific terms which are correct in the presence of polynomial multiplication arithmetic.
Through proper verification:
4. Verification:
After verifying the factor combinations, we find the correct polynomial products that fit precisely.
[tex]\[
8x^2 + 10x - 3 = (2x + 3)(4x - 1)
\][/tex]
Hence, upon full confirmation,
### Conclusion:
The correct factor matching with our trinomial is confirmed:
So, the binomial [tex]\(2x + 3\)[/tex] (Option A) is indeed a suitable factor for [tex]\(8x^2 + 10x - 3\)[/tex].
Thus, the answer is:
[tex]\[ \boxed{A} \][/tex]
