So you start with a fraction you wish to simplify (it could just be a division problem but simplification is the most common practical application, and it makes no difference). First, take the denominator, set it equal to 0, and solve for x, write that value down somewhere. Now write the coefficients of the numerator out left to right (highest power of x on the left, be sure to put a 0 in if there's a missing power). Start with the highest power of x (furthest left), write it again below. Multiply it by the value from the first step, and write that number below the coefficient of the next highest power, then add those two and write the sum below. Multiply the sum by the number from the first step, write the product under the coefficient of the next largest power, and repeat until you've used all of the coefficients. Then, with the results (the lowest line of numbers that was found with the sums) bump all of the powers of x down one (the result under the coefficient of x^4 becomes result*x^3). If the final result value is 0, you have no remainder, and you're done. If it's not, divide it by the original denominator and then you're done.
I hope that helped, I'm not the best at explaining things. It's really exactly what you do in long division but in base x instead of base 10.
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Anwar Al-Awlaki
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