Let's determine which of the multiple-choice options is equivalent to the expression [tex]\((2x - y)(7 - 8x)\)[/tex].
To solve this, we can expand the product of the two binomials step-by-step:
1. Distribute each term in the first binomial ([tex]\(2x - y\)[/tex]) to each term in the second binomial ([tex]\(7 - 8x\)[/tex]):
[tex]\[
(2x - y)(7 - 8x) = 2x \cdot 7 + 2x \cdot (-8x) - y \cdot 7 - y \cdot (-8x)
\][/tex]
2. Calculate each of the products individually:
[tex]\[
2x \cdot 7 = 14x
\][/tex]
[tex]\[
2x \cdot (-8x) = -16x^2
\][/tex]
[tex]\[
-y \cdot 7 = -7y
\][/tex]
[tex]\[
-y \cdot (-8x) = 8xy
\][/tex]
3. Combine all the resulting terms:
[tex]\[
14x - 16x^2 - 7y + 8xy
\][/tex]
4. Reorder the terms to match standard polynomial form:
[tex]\[
-16x^2 + 8xy + 14x - 7y
\][/tex]
Now, let's compare this expanded expression with the given options.
A. [tex]\((14 + 8y)x\)[/tex]:
- This expression is not equivalent to [tex]\(-16x^2 + 8xy + 14x - 7y\)[/tex] since it lacks several terms, such as [tex]\(-16x^2\)[/tex] and [tex]\(-7y\)[/tex].
B. [tex]\(-16x^2 + (14 + 8y)x - 7y\)[/tex]:
- This expression is not equivalent either because the term [tex]\((14 + 8y)x\)[/tex] suggests there is a single multipliable term which isn't the case in our expanded expression.
C. [tex]\(-6x - y + 7\)[/tex]:
- This expression is completely different and does not match our expanded terms.
D. [tex]\(-16x^2 + 22xy - 7y\)[/tex]:
- This term is also incorrect because it shows [tex]\(22xy\)[/tex] instead of [tex]\(8xy\)[/tex].
Given the expanded expression [tex]\(-16x^2 + 8xy + 14x - 7y\)[/tex], we conclude that none of the provided multiple-choice options is exactly equivalent. It appears there might be an error in the options given.
However, based on the best possible scenario, to match the given expanded expression:
- The closest form in structure (though incorrect numerically) is:
[tex]\[
-16 x^2+22 x y-7 y
\][/tex]
Thus, even though none fit correctly based on precise terms given in the question. The correct expansion should be anticipated but there could be a discrepancy in choices provided.
Since choices have predefined given solutions which choose to predefine closest potential, it is:
D. [tex]\(-16x^2 + 22xy - 7y\)[/tex] yet logically correcting exact match require proper option adjustments apparently mistaken in given question formats.
