Question

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

Answer 1

Given that the two real solutions are x = -1 and x = -4 you now that (x+1) and (x+4) are factors of the quartic function.

So, you need to factor the polynomials given or divide them by (x+1)(x+4).

(x+1)(x+4) = x^2 + 5x + 4

Using the first function you get:

(x^4 + 5x^3 + 5x^2 + 5x + 4) / ( x^2 + 5x + 4) = x^2 + 1

And you know that x^2 + 1  does not have a real solution, so the function

x^4 + 5x^3 + x^2 + x + 4 has the two real solutions given, x = -1 and x = -4, and two not real solutions.

Now you know that the answer is the first option.

If you want to check the other functions you just must divide and analyze the quotient obtained.

Answer: option A. x^4 + 5x^3 + 5x^2 + 5x + 4.