What is a quartic function with only the two real zeros given?

x = -4 and x = -1

A. y = x^4 + 5x^3 + 5x^2 + 5x + 4

B. y = x^4 - 5x^3 - 5x^2 - 5x - 4

C. y = -x^4 + 5x^3 + 5x^2 + 5x + 4

D. y = x^4 + 5x^3 + 5x^2 + 5x - 5

If x=-4 and x = -1 are zeros to one or more of the given functions, then dividing them by both x+1 and x+4 should yield a remainder of zero.

I divided x+1 into each of the given functions (I used synthetic division), each yielded a remainder.
So x+1 did not divide into any of them

I suspect either a typo, or the question is flawed (or I made an error in my calculation)

To find the quartic function with the given zeros, you need to use the factored form of a polynomial equation.

The factored form of a quartic equation can be written as (x - a)(x - b)(x - c)(x - d), where a, b, c, and d are the zeros of the function.

In this case, the given zeros are x = -4 and x = -1.

Therefore, the factored form of the quartic function would be (x + 4)(x + 1)(x - a)(x - b).

To find the values of a and b, you can expand the factored form and compare it to the given options.

Expanding (x + 4)(x + 1)(x - a)(x - b) would result in a quartic polynomial with four terms.

By comparing the coefficients of the expanded form to the options, you can determine the correct answer.

Let's expand the factored form and compare the coefficients:

(x + 4)(x + 1)(x - a)(x - b)
= (x^2 + 5x + 4)(x - a)(x - b)
= (x^2 + 5x + 4)(x^2 - (a + b)x + ab)
= x^4 + 5x^3 + 4x^2 - (a + b)x^3 - 5(a + b)x^2 - 4(a + b)x + abx^2 + 5abx + 4ab

Comparing this expanded form to the given options, we can see that option A matches the coefficients:

y = x^4 + 5x^3 + 5x^2 + 5x + 4

Therefore, the correct answer is A.