To determine which expression is equivalent to [tex]\( 2x^2 - 14x + 24 \)[/tex], we need to factor the polynomial completely and then compare it to the given choices.
Given expression:
[tex]\[ 2x^2 - 14x + 24 \][/tex]
We need to determine its factored form. We will compare this factored form with the choices provided.
Let's examine the factored forms provided in the choices:
Option A:
[tex]\[ (2x-12)(x-2) \][/tex]
Let's expand and simplify:
[tex]\[ (2x - 12)(x - 2) = 2x(x - 2) - 12(x - 2) = 2x^2 - 4x -12x + 24 = 2x^2 - 16x + 24 \][/tex]
This is not equal to [tex]\( 2x^2 - 14x + 24 \)[/tex].
Option B:
[tex]\[ 2(x-5)(x-2) \][/tex]
Let's expand and simplify:
[tex]\[ 2(x - 5)(x - 2) = 2[(x - 5)(x - 2)] = 2[x^2 - 2x - 5x + 10] = 2[x^2 - 7x + 10] = 2x^2 - 14x + 20 \][/tex]
This is not equal to [tex]\( 2x^2 - 14x + 24 \)[/tex].
Option C:
[tex]\[ 2(x-3)(x-4) \][/tex]
Let's expand and simplify:
[tex]\[ 2(x - 3)(x - 4) = 2[(x - 3)(x - 4)] = 2[x^2 - 4x - 3x + 12] = 2[x^2 - 7x + 12] = 2x^2 - 14x + 24 \][/tex]
This simplifies to [tex]\( 2x^2 - 14x + 24 \)[/tex].
Option D:
[tex]\[ 2(x-8)(x+3) \][/tex]
Let's expand and simplify:
[tex]\[ 2(x - 8)(x + 3) = 2[(x - 8)(x + 3)] = 2[x^2 + 3x - 8x - 24] = 2[x^2 - 5x - 24] = 2x^2 - 10x - 48 \][/tex]
This is not equal to [tex]\( 2x^2 - 14x + 24 \)[/tex].
After comparing all the options, we find:
The correct expression equivalent to [tex]\( 2x^2 - 14x + 24 \)[/tex] is:
[tex]\[ \boxed{2(x - 3)(x - 4)} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
