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We divide x to the third plus five x minus four by x minus x, Actually I think this is supposed to be

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x squared, let me correct it. So we are gonna to divide, x to the third plus five x minus four by x squared

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minus x plus 1 and just to see the different ways we can rewrite this, we could rewrite this as , x to

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the third plus five x minus four divided by x squared minus x plus 1

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or may be the best way to write in this circumstance is we are going to write algebric long division

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is to write it as x squared minus x plus one

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divided into x to the third plus

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and actually I am going to leave some blank spaces we dont have an x degree an x squared term

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but I am gonna leave some space for it just so that we can align everything in the proper space

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when we actually do the division

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so, x to the third plus nothing to the x squared power plus five x minus four

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so we have a place for the third power, the second power, the first power

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and the zero'th power, so now lets do a little bit of algebraic long division

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lets look at the highest degree term,

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x squared goes into the x to the third, how many times?

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Well it goes into it x times, x to the third divided by x squared

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is equal to x to the three minus two which is equal to x to the one

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which is equal to x, so it goes x times

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I will write the x right over here and we multiply x times this entire thing

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x times, x square is x to the third,

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x times, negative x is negative x squared,

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x times one is , positive x

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and now we want to subtract this whole expression from that whole expression

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and thats the same thing as adding the opposite or

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multiplying each of these terms by negative one and adding into these terms

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so lets do that

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so we have negative x to the third,

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negative one times negative x squared is positive x squared