>>9096922So anyway real talk OP, this is generally what you want to do.
First of all, get your terminology straight. you wrote "ax^3 + bx^2 + cx + d", which is a polynomial. But it's not a polynomial /equation/. For that, set what you wrote equal to zero.
Also recognize that your polynomial involves /one, single variable/. Thus it is a /univariate/ polynomial, as opposed to a /multivariate/ polynomial (where the latter adjective describes more than one variable).
Next, divide through by the leading (manifestly non-zero, so you can divide by it) coefficient a. This leads to a new, /monic/ polynomial, or monic univariate polynomial equation.
Now comes the first real step where you get to do your own work to convince yourself. you make the substitution x = t - b/(3a), which ultimately leads to a new polynomial equation where its next-highest degree term has a zero coefficient, and therefore vanishes. Such a polynomial equation, with a zero-coefficient for its second highest degree, is called a /depressed/ polynomial equation.
When you put all of these terms together, what do you end up with? /A monic, depressed, univariate, polynomial, equation/. I refer to it as an "MDUPE". This is partly autistic, but partly-not. This whole problem is bookkeeping, and you need to keep your books straight and understand clearly exactly what it is that you're working with. that's why I've stressed the above terms in this order.
Your situation is the analogue of the quadratic equation, leading to the quadratic formula. It's just two/three-ish orders of magnitude more complex.
