which second degree polynomial function has a leading coefficient of -1 and root 4 with multiplicity…

Explanation:

Step1: Recall polynomial - root form

If a polynomial has a root (r) with multiplicity (m), then ((x - r)^m) is a factor of the polynomial. Here, the root is (r = 4) with multiplicity (m = 2), so a factor is ((x - 4)^2).

Step2: Consider the leading - coefficient

The polynomial has a leading coefficient of (- 1). So the polynomial (f(x)=-1\times(x - 4)^2).

Step3: Expand the polynomial

Expand ((x - 4)^2) using the formula ((a - b)^2=a^{2}-2ab + b^{2}), where (a=x) and (b = 4). ((x - 4)^2=x^{2}-8x + 16). Then (f(x)=-1\times(x^{2}-8x + 16)=-x^{2}+8x - 16).

Answer:

(f(x)=-x^{2}+8x - 16) (corresponding to the second option)