Simplify ((r+6)/r-1/(r+2))/((r^2+4r+3)/(r^2+2r))

$\frac{\frac{r + 6}{r} - \frac{1}{r + 2}}{\frac{r^{2} + 4 r + 3}{r^{2} + 2 r}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{r + 6}{r} - \frac{1}{r + 2}) \frac{r^{2} + 2 r}{r^{2} + 4 r + 3}$

Step 2

Factor $r$ out of $r^{2} + 2 r$ .

Factor $r$ out of $r^{2}$ .

$(\frac{r + 6}{r} - \frac{1}{r + 2}) \frac{r \cdot r + 2 r}{r^{2} + 4 r + 3}$

Factor $r$ out of $2 r$ .

$(\frac{r + 6}{r} - \frac{1}{r + 2}) \frac{r \cdot r + r \cdot 2}{r^{2} + 4 r + 3}$

Factor $r$ out of $r \cdot r + r \cdot 2$ .

$(\frac{r + 6}{r} - \frac{1}{r + 2}) \frac{r (r + 2)}{r^{2} + 4 r + 3}$

Step 3

Factor $r^{2} + 4 r + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

$(\frac{r + 6}{r} - \frac{1}{r + 2}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 4

To write $\frac{r + 6}{r}$ as a fraction with a common denominator, multiply by $\frac{r + 2}{r + 2}$ .

$(\frac{r + 6}{r} \cdot \frac{r + 2}{r + 2} - \frac{1}{r + 2}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 5

To write $- \frac{1}{r + 2}$ as a fraction with a common denominator, multiply by $\frac{r}{r}$ .

$(\frac{r + 6}{r} \cdot \frac{r + 2}{r + 2} - \frac{1}{r + 2} \cdot \frac{r}{r}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 6

Write each expression with a common denominator of $r (r + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{r + 6}{r}$ by $\frac{r + 2}{r + 2}$ .

$(\frac{(r + 6) (r + 2)}{r (r + 2)} - \frac{1}{r + 2} \cdot \frac{r}{r}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Multiply $\frac{1}{r + 2}$ by $\frac{r}{r}$ .

$(\frac{(r + 6) (r + 2)}{r (r + 2)} - \frac{r}{(r + 2) r}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Reorder the factors of $(r + 2) r$ .

$(\frac{(r + 6) (r + 2)}{r (r + 2)} - \frac{r}{r (r + 2)}) \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 7

Combine the numerators over the common denominator.

$\frac{(r + 6) (r + 2) - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 8

Simplify the numerator.

Expand $(r + 6) (r + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{r (r + 2) + 6 (r + 2) - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Apply the distributive property.

$\frac{r \cdot r + r \cdot 2 + 6 (r + 2) - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Apply the distributive property.

$\frac{r \cdot r + r \cdot 2 + 6 r + 6 \cdot 2 - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

$\frac{r^{2} + r \cdot 2 + 6 r + 6 \cdot 2 - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

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

$\frac{r^{2} + 2 \cdot r + 6 r + 6 \cdot 2 - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Multiply $6$ by $2$ .

$\frac{r^{2} + 2 r + 6 r + 12 - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Add $2 r$ and $6 r$ .

$\frac{r^{2} + 8 r + 12 - r}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Subtract $r$ from $8 r$ .

$\frac{r^{2} + 7 r + 12}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Factor $r^{2} + 7 r + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Write the factored form using these integers.

$\frac{(r + 3) (r + 4)}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 1) (r + 3)}$

Step 9

Simplify terms.

Cancel the common factor of $r + 3$ .

Factor $r + 3$ out of $(r + 1) (r + 3)$ .

$\frac{(r + 3) (r + 4)}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 3) (r + 1)}$

Cancel the common factor.

$\frac{(r + 3) (r + 4)}{r (r + 2)} \cdot \frac{r (r + 2)}{(r + 3) (r + 1)}$

Rewrite the expression.

$\frac{r + 4}{r (r + 2)} \cdot \frac{r (r + 2)}{r + 1}$

Cancel the common factor of $r (r + 2)$ .

Cancel the common factor.

$\frac{r + 4}{r (r + 2)} \cdot \frac{r (r + 2)}{r + 1}$

Rewrite the expression.

$(r + 4) \frac{1}{r + 1}$

Multiply $r + 4$ by $\frac{1}{r + 1}$ .

$\frac{r + 4}{r + 1}$

Simplify ((r+6)/r-1/(r+2))/((r^2+4r+3)/(r^2+2r))

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