Solve frac{r^{2}+3rs+2s^{2}}{r^{2}-2rs-3s^{2}}*r^2-4rs+3s^2/r^2-s^2*r-s/r+2s

$$\frac { r ^ { 2 } + 3 r s + 2 s ^ { 2 } } { r ^ { 2 } - 2 r s - 3 s ^ { 2 } } \times \frac { r ^ { 2 } - 4 r s + 3 s ^ { 2 } } { r ^ { 2 } - s ^ { 2 } } \times \frac { r - s } { r + 2 s }$$

Evaluate

$$\frac{r-s}{r+s}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{r^{2}+3rs+2s^{2}}{r^{2}-2rs-3s^{2}}.

$$\frac{\left(r+s\right)\left(r+2s\right)}{\left(r+s\right)\left(r-3s\right)}\times \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}\times \frac{r-s}{r+2s}$$

Cancel out r+s in both numerator and denominator.

$$\frac{r+2s}{r-3s}\times \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}\times \frac{r-s}{r+2s}$$

Factor the expressions that are not already factored in \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}.

$$\frac{r+2s}{r-3s}\times \frac{\left(r-3s\right)\left(r-s\right)}{\left(r+s\right)\left(r-s\right)}\times \frac{r-s}{r+2s}$$

Cancel out r-s in both numerator and denominator.

$$\frac{r+2s}{r-3s}\times \frac{r-3s}{r+s}\times \frac{r-s}{r+2s}$$

Multiply \frac{r+2s}{r-3s} times \frac{r-3s}{r+s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r+2s\right)\left(r-3s\right)}{\left(r-3s\right)\left(r+s\right)}\times \frac{r-s}{r+2s}$$

Cancel out r-3s in both numerator and denominator.

$$\frac{r+2s}{r+s}\times \frac{r-s}{r+2s}$$

Multiply \frac{r+2s}{r+s} times \frac{r-s}{r+2s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r+2s\right)\left(r-s\right)}{\left(r+s\right)\left(r+2s\right)}$$

Cancel out r+2s in both numerator and denominator.

$$\frac{r-s}{r+s}$$

Expand

$$-\frac{s-r}{r+s}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{r^{2}+3rs+2s^{2}}{r^{2}-2rs-3s^{2}}.

$$\frac{\left(r+s\right)\left(r+2s\right)}{\left(r+s\right)\left(r-3s\right)}\times \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}\times \frac{r-s}{r+2s}$$

Cancel out r+s in both numerator and denominator.

$$\frac{r+2s}{r-3s}\times \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}\times \frac{r-s}{r+2s}$$

Factor the expressions that are not already factored in \frac{r^{2}-4rs+3s^{2}}{r^{2}-s^{2}}.

$$\frac{r+2s}{r-3s}\times \frac{\left(r-3s\right)\left(r-s\right)}{\left(r+s\right)\left(r-s\right)}\times \frac{r-s}{r+2s}$$

Cancel out r-s in both numerator and denominator.

$$\frac{r+2s}{r-3s}\times \frac{r-3s}{r+s}\times \frac{r-s}{r+2s}$$

Multiply \frac{r+2s}{r-3s} times \frac{r-3s}{r+s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r+2s\right)\left(r-3s\right)}{\left(r-3s\right)\left(r+s\right)}\times \frac{r-s}{r+2s}$$

Cancel out r-3s in both numerator and denominator.

$$\frac{r+2s}{r+s}\times \frac{r-s}{r+2s}$$

Multiply \frac{r+2s}{r+s} times \frac{r-s}{r+2s} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r+2s\right)\left(r-s\right)}{\left(r+s\right)\left(r+2s\right)}$$

Cancel out r+2s in both numerator and denominator.

$$\frac{r-s}{r+s}$$