You can write such a cubic polynomial in terms of its roots using the factor theorem:
(x - (-1)) (x - 3) (x - (-5)) = (x + 1) (x - 3) (x + 5)
Now just expand this:
(x + 1) (x - 3) (x + 5) = x ³ + (1 - 3 + 5) x ² + (-3 + 5 - 15) x - 15
… = x ³ + 3x ² - 13x - 15
