First we set up the long division problem. We make sure the terms of the polynomials are written in descending order by exponent, and we leave space in the dividend for the x4 and x3 terms, even though we don't have any. It's like leaving a place at the table for your dad, even though he's currently in Baltimore on business. Dressing up your stuffed rabbit in a fedora and glasses and seating him in your dad's chair might be going too far, though.
We'll work our way through the long division. First, figure out how many times the first term of the divisor goes into the first term of the dividend, perform the necessary subtraction, and bring down the remaining terms of the polynomial:
Now we see how many times the first term of the divisor goes into the first term of our new polynomial, and carry out the appropriate subtraction:
The final answer is .
Check the answer to the above example by multiplying (2x4 + 3) by (4x2 + x). You should get 8x6 + 12x2 + 2x5 + 3x. If you don't, go back to square one. Collect $200.