Divide ((x+1)^(2n)+4(x+1)^(2n+1)+6(x+1)^6-11)/(x(x+2))

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 {(x + 1)}^{6} - 11}{x (x + 2)}$

Step 1

Simplify the numerator.

Use the Binomial Theorem.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} \cdot 1 + 15 x^{4} \cdot 1^{2} + 20 x^{3} \cdot 1^{3} + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

Simplify each term.

Multiply $6$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} \cdot 1^{2} + 20 x^{3} \cdot 1^{3} + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

One to any power is one.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} \cdot 1 + 20 x^{3} \cdot 1^{3} + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

Multiply $15$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} \cdot 1^{3} + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

One to any power is one.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} \cdot 1 + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

Multiply $20$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} \cdot 1^{4} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

One to any power is one.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} \cdot 1 + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

Multiply $15$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x \cdot 1^{5} + 1^{6}) - 11}{x (x + 2)}$

One to any power is one.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x \cdot 1 + 1^{6}) - 11}{x (x + 2)}$

Multiply $6$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1^{6}) - 11}{x (x + 2)}$

One to any power is one.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1) - 11}{x (x + 2)}$

Apply the distributive property.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 6 (6 x^{5}) + 6 (15 x^{4}) + 6 (20 x^{3}) + 6 (15 x^{2}) + 6 (6 x) + 6 \cdot 1 - 11}{x (x + 2)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 6 (15 x^{4}) + 6 (20 x^{3}) + 6 (15 x^{2}) + 6 (6 x) + 6 \cdot 1 - 11}{x (x + 2)}$

Multiply $15$ by $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 6 (20 x^{3}) + 6 (15 x^{2}) + 6 (6 x) + 6 \cdot 1 - 11}{x (x + 2)}$

Multiply $20$ by $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 6 (15 x^{2}) + 6 (6 x) + 6 \cdot 1 - 11}{x (x + 2)}$

Multiply $15$ by $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 6 (6 x) + 6 \cdot 1 - 11}{x (x + 2)}$

Multiply $6$ by $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x + 6 \cdot 1 - 11}{x (x + 2)}$

Multiply $6$ by $1$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x + 6 - 11}{x (x + 2)}$

Subtract $11$ from $6$ .

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x - 5}{x (x + 2)}$

Step 2

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x - 5}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 3

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2} + 36 x}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 4

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3} + 90 x^{2}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 5

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4} + 120 x^{3}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 6

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5} + 90 x^{4}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 7

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6} + 36 x^{5}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6}}{x (x + 2)} + \frac{36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 8

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1} + 6 x^{6}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{6}}{x (x + 2)} + \frac{36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 9

Split the fraction $\frac{{(x + 1)}^{2 n} + 4 {(x + 1)}^{2 n + 1}}{x (x + 2)}$ into two fractions.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{6}}{x (x + 2)} + \frac{36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 10

Cancel the common factor of $x^{6}$ and $x$ .

Factor $x$ out of $6 x^{6}$ .

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{x (6 x^{5})}{x (x + 2)} + \frac{36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{5}}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 11

Cancel the common factor of $x^{5}$ and $x$ .

Factor $x$ out of $36 x^{5}$ .

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{x (36 x^{4})}{x (x + 2)} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{4}}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 12

Cancel the common factor of $x^{4}$ and $x$ .

Factor $x$ out of $90 x^{4}$ .

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{x (90 x^{3})}{x (x + 2)} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{3}}{x + 2} + \frac{120 x^{3}}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 13

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $120 x^{3}$ .

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{3}}{x + 2} + \frac{x (120 x^{2})}{x (x + 2)} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{3}}{x + 2} + \frac{120 x^{2}}{x + 2} + \frac{90 x^{2}}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 14

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $90 x^{2}$ .

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{3}}{x + 2} + \frac{120 x^{2}}{x + 2} + \frac{x (90 x)}{x (x + 2)} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

$\frac{{(x + 1)}^{2 n}}{x (x + 2)} + \frac{4 {(x + 1)}^{2 n + 1}}{x (x + 2)} + \frac{6 x^{5}}{x + 2} + \frac{36 x^{4}}{x + 2} + \frac{90 x^{3}}{x + 2} + \frac{120 x^{2}}{x + 2} + \frac{90 x}{x + 2} + \frac{36 x}{x (x + 2)} + \frac{- 5}{x (x + 2)}$

Step 15

Cancel the common factor of $x$ .

Cancel the common factor.

Rewrite the expression.

Step 16

Move the negative in front of the fraction.

Divide ((x+1)^(2n)+4(x+1)^(2n+1)+6(x+1)^6-11)/(x(x+2))

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