Explanation:
p(x) = ax³ + 3bx² + 3cx + d
Roots are α−β, α, and α+β.
The sum of the roots of a cubic y = ax³ + bx² + cx + d is -b/a. Therefore:
α−β + α + α+β = -3b/a
3α = -3b/a
α = -b/a
α is a root, so p(α) = 0.
0 = a(α)³ + 3b(α)² + 3c(α) + d
Substitute α = -b/a:
0 = a(-b/a)³ + 3b(-b/a)² + 3c(-b/a) + d
0 = -(b³/a²) + 3b³/a² − 3bc/a + d
0 = -b³ + 3b³ − 3abc + a²d
0 = 2b³ − 3abc + a²d
3abc = 2b³ + a²d