How to solve a 4th degree polynomial?

Question

I am feeling difficulty to find the roots of this 4th degree polynomial:

$3x^4+26x^3+77x^2+84x+24=0$

Factorization methods have been tried.

Answer 1

There are two real roots, and two complex roots to your polynomial. Even the real roots are rather complicated. They do not lend themselves to any sort of nice factoring.

You can estimate the roots by graphing the polynomial, and zeroing in at the intersection points of the graph with the $x$-axis,

or use graphing software like Wolfram Alpha to give exact forms for the solutions.

Answer 2

Once you have seen the plot of the function and noticed that it is well conditioned in the area of the roots, you can apply a Newton method which writes
x_new = x_old - f(x_old) / f'(x_old)
Suppose we start with x_old = -1.5, the successive iterates will then be -1.6094 and -1.6097; you can continue this process until you meet the required accuracy.
Suppose we start with x_old = -0.5, the successive iterates will then be -0.4275, -0.4348 and -0.4349; you can continue this process until you meet the required accuracy.
The approximate solutions are -1.60971770887 and -0.434922318772; these are the numerical values of the nice expressions posted by Oliver Bel