Factor (x^2+3x+2)/(x(x+2)^3(x+1))-1/(x(x+2)(x+5))

$\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)} - \frac{1}{x (x + 2) (x + 5)}$

Step 1

To write $\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)} \cdot \frac{x + 5}{x + 5} - \frac{1}{x (x + 2) (x + 5)}$

Step 2

To write $- \frac{1}{x (x + 2) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{(x + 1) {(x + 2)}^{2}}{(x + 1) {(x + 2)}^{2}}$ .

$\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)} \cdot \frac{x + 5}{x + 5} - \frac{1}{x (x + 2) (x + 5)} \cdot \frac{(x + 1) {(x + 2)}^{2}}{(x + 1) {(x + 2)}^{2}}$

Step 3

Write each expression with a common denominator of $x (x + 1) (x + 5) {(x + 2)}^{3}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{1}{x (x + 2) (x + 5)} \cdot \frac{(x + 1) {(x + 2)}^{2}}{(x + 1) {(x + 2)}^{2}}$

Multiply $\frac{1}{x (x + 2) (x + 5)}$ by $\frac{(x + 1) {(x + 2)}^{2}}{(x + 1) {(x + 2)}^{2}}$ .

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{(x + 1) {(x + 2)}^{2}}{x (x + 2) (x + 5) ((x + 1) {(x + 2)}^{2})}$

Raise $x + 2$ to the power of $1$ .

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{(x + 1) {(x + 2)}^{2}}{x ({(x + 2)}^{2} {(x + 2)}^{1}) (x + 5) (x + 1)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{(x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{2 + 1} (x + 5) (x + 1)}$

Add $2$ and $1$ .

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{(x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 5) (x + 1)}$

Reorder the factors of $x {(x + 2)}^{3} (x + 5) (x + 1)$ .

$\frac{(x^{2} + 3 x + 2) (x + 5)}{x {(x + 2)}^{3} (x + 1) (x + 5)} - \frac{(x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x^{2} + 3 x + 2) (x + 5) - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Step 5

Rewrite $\frac{(x^{2} + 3 x + 2) (x + 5) - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$ in a factored form.

Expand $(x^{2} + 3 x + 2) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} x + x^{2} \cdot 5 + 3 x \cdot x + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{x^{2} x^{1} + x^{2} \cdot 5 + 3 x \cdot x + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2 + 1} + x^{2} \cdot 5 + 3 x \cdot x + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $2$ and $1$ .

$\frac{x^{3} + x^{2} \cdot 5 + 3 x \cdot x + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Move $5$ to the left of $x^{2}$ .

$\frac{x^{3} + 5 x^{2} + 3 x \cdot x + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x^{3} + 5 x^{2} + 3 (x \cdot x) + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $x$ by $x$ .

$\frac{x^{3} + 5 x^{2} + 3 x^{2} + 3 x \cdot 5 + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $5$ by $3$ .

$\frac{x^{3} + 5 x^{2} + 3 x^{2} + 15 x + 2 x + 2 \cdot 5 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $2$ by $5$ .

$\frac{x^{3} + 5 x^{2} + 3 x^{2} + 15 x + 2 x + 10 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $5 x^{2}$ and $3 x^{2}$ .

$\frac{x^{3} + 8 x^{2} + 15 x + 2 x + 10 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $15 x$ and $2 x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - (x + 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Apply the distributive property.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1 \cdot 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $- 1$ by $1$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) {(x + 2)}^{2}}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) ((x + 2) (x + 2))}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x (x + 2) + 2 (x + 2))}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Apply the distributive property.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x \cdot x + x \cdot 2 + 2 (x + 2))}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Apply the distributive property.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Move $2$ to the left of $x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x^{2} + 2 x + 2 x + 2 \cdot 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $2$ by $2$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x^{2} + 2 x + 2 x + 4)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $2 x$ and $2 x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 + (- x - 1) (x^{2} + 4 x + 4)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Expand $(- x - 1) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x \cdot x^{2} - x (4 x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - (x^{2} x) - x (4 x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - (x^{2} x^{1}) - x (4 x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{2 + 1} - x (4 x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $2$ and $1$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - x (4 x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 1 \cdot 4 x \cdot x - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 1 \cdot 4 (x \cdot x) - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $x$ by $x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 1 \cdot 4 x^{2} - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $- 1$ by $4$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 4 x^{2} - x \cdot 4 - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $4$ by $- 1$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 4 x^{2} - 4 x - 1 x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 4 x^{2} - 4 x - x^{2} - 1 (4 x) - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $4$ by $- 1$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 4 x^{2} - 4 x - x^{2} - 4 x - 1 \cdot 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Multiply $- 1$ by $4$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 4 x^{2} - 4 x - x^{2} - 4 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $x^{2}$ from $- 4 x^{2}$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 5 x^{2} - 4 x - 4 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $4 x$ from $- 4 x$ .

$\frac{x^{3} + 8 x^{2} + 17 x + 10 - x^{3} - 5 x^{2} - 8 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $x^{3}$ from $x^{3}$ .

$\frac{8 x^{2} + 17 x + 10 + 0 - 5 x^{2} - 8 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Add $8 x^{2}$ and $0$ .

$\frac{8 x^{2} + 17 x + 10 - 5 x^{2} - 8 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $5 x^{2}$ from $8 x^{2}$ .

$\frac{3 x^{2} + 17 x + 10 - 8 x - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $8 x$ from $17 x$ .

$\frac{3 x^{2} + 9 x + 10 - 4}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Subtract $4$ from $10$ .

$\frac{3 x^{2} + 9 x + 6}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Rewrite $3 x^{2} + 9 x + 6$ in a factored form.

Factor $3$ out of $3 x^{2} + 9 x + 6$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) + 9 x + 6}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Factor $3$ out of $9 x$ .

$\frac{3 (x^{2}) + 3 (3 x) + 6}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Factor $3$ out of $6$ .

$\frac{3 x^{2} + 3 (3 x) + 3 \cdot 2}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Factor $3$ out of $3 x^{2} + 3 (3 x)$ .

$\frac{3 (x^{2} + 3 x) + 3 \cdot 2}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Factor $3$ out of $3 (x^{2} + 3 x) + 3 \cdot 2$ .

$\frac{3 (x^{2} + 3 x + 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Factor.

Factor $x^{2} + 3 x + 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$\frac{3 ((x + 1) (x + 2))}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Remove unnecessary parentheses.

$\frac{3 (x + 1) (x + 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Reduce the expression $\frac{3 (x + 1) (x + 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$ by cancelling the common factors.

Cancel the common factor.

$\frac{3 (x + 1) (x + 2)}{x {(x + 2)}^{3} (x + 1) (x + 5)}$

Rewrite the expression.

$\frac{3 (x + 2)}{x {(x + 2)}^{3} (x + 5)}$

Step 6

Reduce the expression $\frac{3 (x + 2)}{x {(x + 2)}^{3} (x + 5)}$ by cancelling the common factors.

Factor $x + 2$ out of $3 (x + 2)$ .

$\frac{(x + 2) \cdot 3}{x {(x + 2)}^{3} (x + 5)}$

Factor $x + 2$ out of $x {(x + 2)}^{3} (x + 5)$ .

$\frac{(x + 2) \cdot 3}{(x + 2) (x {(x + 2)}^{2} (x + 5))}$

Cancel the common factor.

$\frac{(x + 2) \cdot 3}{(x + 2) (x {(x + 2)}^{2} (x + 5))}$

Rewrite the expression.

$\frac{3}{x {(x + 2)}^{2} (x + 5)}$

Factor (x^2+3x+2)/(x(x+2)^3(x+1))-1/(x(x+2)(x+5))

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