To determine which of the given binomials is a factor of the trinomial [tex]\( x^2 + 2x - 48 \)[/tex], we need to factor the trinomial.
### Step-by-Step Factoring Process:
1. Identify the Trinomial:
The trinomial is [tex]\( x^2 + 2x - 48 \)[/tex].
2. Look for a Pair of Factors:
We need to find two numbers that multiply to the constant term [tex]\(-48\)[/tex] and add up to the coefficient of the [tex]\( x \)[/tex] term, which is [tex]\( 2 \)[/tex].
- Let's list the pairs of factors of [tex]\(-48\)[/tex]:
[tex]\[
(-1, 48), (1, -48), (-2, 24), (2, -24), (-3, 16), (3, -16), (-4, 12), (4, -12), (-6, 8), (6, -8)
\][/tex]
3. Find the Correct Pair:
We need the pair whose sum is [tex]\( 2 \)[/tex]. Checking the pairs:
- [tex]\((-6, 8)\)[/tex]: The sum is [tex]\(-6 + 8 = 2\)[/tex], which matches the [tex]\( x \)[/tex] coefficient.
4. Write the Binomial Factors:
Now that we have the pair [tex]\((-6, 8)\)[/tex], we can write the trinomial as:
[tex]\[
x^2 + 2x - 48 = (x - 6)(x + 8)
\][/tex]
### Verify the Factorization:
To ensure our factorization is correct, we can expand [tex]\((x - 6)(x + 8)\)[/tex]:
[tex]\[
(x - 6)(x + 8) = x^2 + 8x - 6x - 48 = x^2 + 2x - 48
\][/tex]
The expansion matches our original trinomial.
### Conclusion:
The trinomial [tex]\( x^2 + 2x - 48 \)[/tex] factors into [tex]\((x - 6)(x + 8)\)[/tex].
Therefore, the binomials [tex]\( x + 8 \)[/tex] and [tex]\( x - 6 \)[/tex] are factors. Among the given options, the factor is:
[tex]\[ \boxed{x + 8} \][/tex]
Because [tex]\( x - 6 \)[/tex] was not among the options, the correct answer from the given choices is option:
[tex]\[ \boxed{A. x + 8} \][/tex]
