To factor the quadratic expression [tex]\(2x^2 - 10x - 48\)[/tex] completely, we can follow these steps:
1. Identify the quadratic expression: The given expression is [tex]\(2x^2 - 10x - 48\)[/tex].
2. Find the factors that multiply to give [tex]\(2 \cdot -48 = -96\)[/tex] and add to give the middle coefficient, which is [tex]\(-10\)[/tex].
- The factors of [tex]\(-96\)[/tex] that add up to [tex]\(-10\)[/tex] are [tex]\(6\)[/tex] and [tex]\(-16\)[/tex].
3. Rewrite the middle term using these factors:
[tex]\[
2x^2 - 10x - 48 = 2x^2 + 6x - 16x - 48
\][/tex]
4. Group the terms to factor by grouping:
[tex]\[
(2x^2 + 6x) + (-16x - 48)
\][/tex]
5. Factor out the greatest common factor (GCF) from each group:
[tex]\[
2x(x + 3) - 16(x + 3)
\][/tex]
6. Factor out the common binomial [tex]\(x + 3\)[/tex]:
[tex]\[
(2x - 16)(x + 3)
\][/tex]
7. Factor out the constant [tex]\(2\)[/tex] from [tex]\((2x - 16)\)[/tex]:
[tex]\[
2(x - 8)(x + 3)
\][/tex]
So, the completely factored form of [tex]\(2x^2 - 10x - 48\)[/tex] is [tex]\(2(x - 8)(x + 3)\)[/tex].
Thus, the correct answer is [tex]\(\boxed{2(x + 3)(x - 8)}\)[/tex].
