Question

Find a quartic function with the given x -values as its only real zeros. x=-4 and x=-1 .

Answer 1

If you are emphasizing on the fact that these are its only real roots, then maybe the function has any other 2 imaginary roots, so there wouldn't be any specific equation that you're looking for.

However if you mean that these real roots are the only roots of this function, meaning that they are double roots, we can argue that:

`f(x) = (x + 4) {}^(2) (x + 1) {}^(2) \\ f(x) = (x {}^(2) + 8x + 16)(x {}^(2) + 2x + 1) \\ expand \: by \: multiplying \: each \: term \: by \: the \: other \: 3 \\ f(x) = x {}^(4) + 10x {}^(3) + 33x {}^(2) + 40x + 16`