Consider the polynomial f(x) = 1 + 2x + 3x² + 4x³. Let s be the sum of all distinct real roots of f(x) and let t = |s|. The real number t lies in the interval (-1, -1/2), (-1/4, 0), (-1/2, 1), (0, 1/4)

f(x) = 1 + 2x + 3x² + 4x³
f'(x) = 2 + 6x + 12x²
f'(x) = 2(1 + 3x + 6x²)
D = b² - 4ac (ignoring the constant multiplier 2).
D = 9 - 4 × 6 = -15
This means it is always an increasing function. We can also conclude that f(x) has only one real root.