No, you wouldn't. Or, at least, I wouldn't. If I hadn't noticed that the divisor and dividend are exactly the same, I would think, looking at the highest powers just as in dividing numbers you look at the first few digits to get an estimate of a quotient. x^4 divides into x^4 once so the "quotient" is 1, then we subtract 1(x^4- x^4+ x- 1) from x^4- x^3+ x- 1 and see that my remainder is 0. Problem finished!
No, you don't understand it. You never "divide by the remainder".Then take x^3- x^2+ x- 1 to the next line and divide by the remainder (-x^2+2x-1).
I understand it for small ones like that, but when it's larger ones like x^10 or where the coefficient of x isn't 1 I can't do them...