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My textbook was simplifying a power series. I was able to follow their work until the last line.

Where did the "$-2$" come from?

Why were they able to cancel out one of the $(n+r)'s$, $C_n$'s, and $x^{n+r-1}$'s?

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They didn't show some simplification. In the line above the last line, look at the first two terms. You can factor out the entire second term and get: $$3(n+r-1) + 1$$

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Hint: We have

$$3(n+r)\cdot (n+r-1)+(n+r)$$

Now we factor out $(n+r)$:

$$(n+r)\cdot (3\cdot (n+r-1)+1)=(n+r)\cdot (3n+3r\underbrace{-3+1}_{-2})=(n+r)\cdot (3n+3r-2)$$

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