Step 1: How It Works
Convince yourself that f(a)=f'(a)(a-b). f'(a) is the derivative of the function at point a. It's the slope of the function at a. If you multiply this by a certain number (a-b), b is undefined, it will equal the function at that point. If we know the value of the function at point a, and the derivative of the function at point a, then we can rewrite the equation in terms of b (the undefined point on the x-axis.) Rewritten as such:
b=a-f(a)/f'(a). Using this equation, we can use any point on the x-axis, find the next point, then use that one, and repeat until we hit one of the zeroes.
Step 2: Let's Find the Roots!
First things first, graph your function. You can use a spreadsheet program, your calculator, whatever works for you. Make a rough estimate as to where the zeroes are. Looking at the graph below, I'm going to estimate that there are zeroes at -2, -1.2, 1.2, and 3. Let's see how accurate my guess is.
Now's a good time to open up your spreadsheet program.
Step 3: Let Excel Do the Work
In cell A1 put in your guess for one of the zeroes. In cell B2 put in your equation as a function of cell A1 (shown in the picture.) In cell C1 put in your derivative as a function of A1.
In cell A2 type in "=A1-B1/C1" this is the equation I showed you a couple steps earlier. Now drag all three cells down 10 or so spaces. The last cell in column A should be your root. To check your answer, make sure that the last cell in column A matches the next to last cell. If any of the numbers are different, drag all three cells down even further. Congrats, there's the first root.
Step 4: Find the Other Roots
That's how Newton's method works.