The equation $x(ax+b) = c$ is valid but does not help. There is a general fact that $AB = 0$ implies $A = 0$ or $B=0$ and this allows us to solve a product expression by reducing it to easier equations. So we need 0 on the right hand side of the product to be useful.
e.g. we want to rewrite your equation $ax^2 + bx - c = 0$ as $a(x+\alpha)(x+\beta) = 0$.
In order to find $\alpha, \beta$ to do this, note that we ensured the quadritic term is already OK: $ax^2$ in both. The linear term is $a(\alpha+\beta) = b$ and the constant term is $a\alpha\beta = -c$. So you need to find $\alpha$ and $\beta$ with known sum $\frac{b}{a}$ and known product $\frac{-c}{a}$, and this can sometimes be seen by inspection for concrete $a,b$ and $c$.