All you need to do is apply the Newton-Raphson method, generally described by the equation:
$\displaystyle x _{n + 1} = x_{n} - \frac{f(x_{n})}{f\prime(x_{n})}$
So in your case:
$\displaystyle x_{n + 1} = x_{n} - \frac{x_{n}^3 + x_{n}^2 + x_{n} - 2}{3x_{n}^2 +2x_{n} +1}$
with an initial guess of 1.
You will probably want to use a calculator to evaluate that equation. If you don't have one, there are many on the internet and Google works as a scientific calculator as well, provided you type the equation in correctly bracketed.
If you use a TI83/TI84 then you can avoid to type a lot of values and operation signs (with the risk of errors).
1. Store the term of the function at Y1.
Store the derivation of the function at Y2
2. Type the start value + ENTER. (in your case 1 ENTER)
3. Now type sinewave85's formula but use the ANS-key instead of x:
ANS - Y1(ANS)/Y2(ANS)
and press ENTER. You'll get the next x-value which is automatically written into ANS. So you only have to press the ENTER-key until two consecutive values are equal.
4. See attachment.