How Long Division Works

You already know long division with numbers. To divide 97 by 4, divide 9 by 4 to get 2, multiply 2 by 4 to get 8, subtract to leave 1, then bring down the 7 to make 17. Repeat: 17 divided by 4 is 4, 4 times 4 is 16, and 17 minus 16 leaves 1. Nothing is left to bring down, so the quotient is 24 and the remainder is 1.

Polynomials divide the same way, with terms in place of digits. To divide x^2 + 3x + 5 by x + 1, divide the leading terms: x^2 over x is x. Multiply: x(x + 1) is x^2 + x. Subtract it from x^2 + 3x to leave 2x, and bring down the 5 to make 2x + 5. Repeat: 2x over x is 2, 2(x + 1) is 2x + 2, and subtracting leaves 3.

You stop when what is left has a lower degree than the divisor: 3 has no x, and x + 1 does. The answer reads quotient plus remainder over divisor, just as 97 over 4 is 24 plus 1 over 4:

(x^2 + 3x + 5)/(x + 1) = x + 2 + 3/(x + 1)

It checks the same way too: (x + 1)(x + 2) + 3 gives back x^2 + 3x + 5.