If a polynomial has a zero or root at x = a, then it has a factor (x - a). So we can simply build the factors for each root, and multiply them together:
f(x) = (x + 4)(x - 0)(x - 1)
f(x) = x(x^2 +3x - 4)
f(x) = x^3 + 3x^2 - 4x
You can check that x = -4, 0 and 1 all give f(x) = 0.
Note that we're just doing the reverse of what we usually do:
Polynomial f(x) = x^n + cx^n-1 ... -> Find factors f(x) = (x + a)(x + b)... -> Get zeros x = -a, x = -b...
The tricky step in this process is almost always getting the factors from a polynomial. But working backwards, it's much easier! You get the factors from the zeros, then just multiply them together :-)