What is expand ((x+2)x!+2(x+1)!)/(x!(x+1)!) ?

The solution to expand ((x+2)x!+2(x+1)!)/(x!(x+1)!) is x/((x+1)!)+2/((x+1)!)+2/(x!)

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expand ((x+2)x!+2(x+1)!)/(x!(x+1)!)

expand

\frac{( x + 2 ) x ! + 2 ( x + 1 ) !}{x ! ( x + 1 ) !}

\frac{x}{( x + 1 ) !} + \frac{2}{( x + 1 ) !} + \frac{2}{x !}

Solution steps

\frac{( x + 2 ) x ! + 2 ( x + 1 ) !}{x ! ( x + 1 ) !}

Expand

(x + 2) x! + 2 (x + 1)! : xx! + 2 x! + 2 (x + 1)!

= \frac{xx ! + 2 x ! + 2 ( x + 1 ) !}{x ! ( x + 1 ) !}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{xx ! + 2 x ! + 2 ( x + 1 ) !}{x ! ( x + 1 ) !} = \frac{xx !}{x ! ( x + 1 ) !} + \frac{2 x !}{x ! ( x + 1 ) !} + \frac{2 ( x + 1 ) !}{x ! ( x + 1 ) !}

= \frac{xx !}{x ! ( x + 1 ) !} + \frac{2 x !}{x ! ( x + 1 ) !} + \frac{2 ( x + 1 ) !}{x ! ( x + 1 ) !}

Cancel

\frac{xx !}{x ! ( x + 1 ) !} : \frac{x}{( x + 1 ) !}

= \frac{x}{( x + 1 ) !} + \frac{2 x !}{x ! ( x + 1 ) !} + \frac{2 ( x + 1 ) !}{x ! ( x + 1 ) !}

Cancel

\frac{2 x !}{x ! ( x + 1 ) !} : \frac{2}{( x + 1 ) !}

= \frac{x}{( x + 1 ) !} + \frac{2}{( x + 1 ) !} + \frac{2 ( x + 1 ) !}{x ! ( x + 1 ) !}

Cancel

\frac{2 ( x + 1 ) !}{x ! ( x + 1 ) !} : \frac{2}{x !}

= \frac{x}{( x + 1 ) !} + \frac{2}{( x + 1 ) !} + \frac{2}{x !}

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\frac{2 x ^{3} + 3 x ^{2} - 15 x - 8}{( x + 2 ) ( x ^{3} - 3 x + 2 )}

What is expand ((x+2)x!+2(x+1)!)/(x!(x+1)!) ?
The solution to expand ((x+2)x!+2(x+1)!)/(x!(x+1)!) is x/((x+1)!)+2/((x+1)!)+2/(x!)