To determine which expression is equivalent to [tex]\(2x^2 - 14x + 24\)[/tex], we can try to factorize it.
Let's factorize the quadratic expression:
[tex]\[2x^2 - 14x + 24\][/tex]
We need to find two numbers that multiply to [tex]\(2 \times 24 = 48\)[/tex] and add up to [tex]\(-14\)[/tex].
The pairs that multiply to 48 are:
- [tex]\(1 \times 48\)[/tex]
- [tex]\(2 \times 24\)[/tex]
- [tex]\(3 \times 16\)[/tex]
- [tex]\(4 \times 12\)[/tex]
- [tex]\(6 \times 8\)[/tex]
When considering their sums and differences, none of these pairs add up to [tex]\(-14\)[/tex] except for [tex]\((-6)\)[/tex] and [tex]\((-8)\)[/tex]:
[tex]\(-6 \times -8 = 48\)[/tex]
[tex]\(-6 + -8 = -14\)[/tex]
Hence, we split the middle term and factor by grouping:
[tex]\[2x^2 - 6x - 8x + 24\][/tex]
Now, group the terms:
[tex]\[2x(x - 3) - 8(x - 3)\][/tex]
This can be factored as:
[tex]\[(2x - 8)(x - 3)\][/tex]
But we see [tex]\(2x - 8\)[/tex] can be further factored as [tex]\(2(x - 4)\)[/tex]:
[tex]\[2(x - 4)(x - 3)\][/tex]
Therefore, the factored form of [tex]\(2x^2 - 14x + 24\)[/tex] is:
[tex]\[2(x - 3)(x - 4)\][/tex]
Thus, the correct answer is:
D. [tex]\(2(x-3)(x-4)\)[/tex]
