Answer:
Choice C. ([tex]-80\, x^4[/tex]).
Step-by-step explanation:
The polynomial here is written as a binomial of degree 5. When this binomial is expanded (not recommended,) there would be six terms with power ranging from [tex]0[/tex] to [tex]5[/tex].
The choices seem to imply that the terms are sorted in decreasing order based on their powers. The first term would be of power five while the fourth would be of power four.
Consider a generic binomial in the form [tex](a\, x + b)^n[/tex]. By the binomial theorem, when the binomial is expanded, the term with power [tex]k[/tex] would be in the form
[tex]\displaystyle {n \choose k}\, a^k\, b^{n - k}\, x^\, k[/tex].
That's the same as
[tex]\displaystyle \frac{n!}{k!(n - k)!}\, a^k\, b^{n - k}\, x^\, k[/tex].
In this question,.
Also, [tex]k = 4[/tex] for the second term.
Hence, that term would be in the form
[tex]\begin{aligned} &\;\frac{5!}{4!\, (5 - 4)!} \, 2^4 \, (-1)^{5 - 4} \, x^4\\ =&\; \frac{5!}{4!\times 1!} \, 2^4 \, (-1)^{1} \, x^4 \\ =&\;\frac{5 \times 4 \times 3 \times 2\times 1}{4 \times 3 \times 2\times 1 \times 1} \times 16 \times (-1) \, x^4 \\ =&\; 5 \times 16\times (-1)\, x \\ =&\; - 80\, x \end{aligned}[/tex].
