Take into account that for a polynomial with zero x=a, one of its factors is (x - a), then, for zeros z=-3,0,3,4, you have
f(x) = (x - (-3))(x - 0)(x - 3)(x - 4)
expand the previous factors:
f(x) = (x + 3)(x)(x - 3)(x - 4)
= x(x + 3)(x - 3)(x - 4) multiply first two factors and last two factors
= (x² + 3x)(x² - 4x - 3x + 12)
= (x² + 3x)(x² - 7x + 12)
= x⁴ - 7x³ + 12x² + 3x³ - 21x² + 36x
= x⁴ - 4x³ - 9x² + 36x
Hence, the quartic polynomial is x⁴ - 4x³ - 9x² + 36x