Main Motivation behind the use of identity of $(a+b)^3$ ,so as to solve cubic equations using Cardons Method

Question

Whats so special about the identity $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ which gave Cardano the clever idea to invent a method to solve cubic depressed equations?

Answer 1

Assume that you have a depressed cubic equation, such as

$$x^3 + bx + c = 0.$$

After (apparent) centuries, Cardano (or somebody) got clever. They reasoned that you could set

$$x = (s + t)$$

and add a second constraint that $3st = -b.$

This meant that

$$s^3 + t^3 + 3st(x) + bx + c = 0. \tag1 $$

However, because of the artificial but valid constraint that $3st = -b$, equation (1) above implies that

$$s^3 + t^3 + c = 0. \tag2 $$

That is, because of this artifical constraint, the $x$ term has vanished.

Equation (2) above implies that

$$s^3 + \left[\frac{-b}{3s}\right]^3 + c = 0. \tag3 $$

In (3) above, multiplying each term by $(s^3)$ converts the equation into a quadratic equation in $(s^3).$