Solve frac{r^{2}-r}{r^{2}-2r-3}*r^2+2r+1/r^2+4rdivr^2-3r-4/r^2-16

$$\frac { r ^ { 2 } - r } { r ^ { 2 } - 2 r - 3 } \cdot \frac { r ^ { 2 } + 2 r + 1 } { r ^ { 2 } + 4 r } \div \frac { r ^ { 2 } - 3 r - 4 } { r ^ { 2 } - 16 }$$

Evaluate

$$\frac{r-1}{r-3}$$

Short Solution Steps

Multiply \frac{r^{2}-r}{r^{2}-2r-3} times \frac{r^{2}+2r+1}{r^{2}+4r} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)}}{\frac{r^{2}-3r-4}{r^{2}-16}}$$

Factor the expressions that are not already factored in \frac{r^{2}-3r-4}{r^{2}-16}.

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

Cancel out r-4 in both numerator and denominator.

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

Divide \frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)} by \frac{r+1}{r+4} by multiplying \frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)} by the reciprocal of \frac{r+1}{r+4}.

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

Factor the expressions that are not already factored.

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

Cancel out r\left(r+4\right)\left(r+1\right)^{2} in both numerator and denominator.

$$\frac{r-1}{r-3}$$

Expand

$$\frac{r-1}{r-3}$$

Short Solution Steps

Multiply \frac{r^{2}-r}{r^{2}-2r-3} times \frac{r^{2}+2r+1}{r^{2}+4r} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)}}{\frac{r^{2}-3r-4}{r^{2}-16}}$$

Factor the expressions that are not already factored in \frac{r^{2}-3r-4}{r^{2}-16}.

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

Cancel out r-4 in both numerator and denominator.

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

Divide \frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)} by \frac{r+1}{r+4} by multiplying \frac{\left(r^{2}-r\right)\left(r^{2}+2r+1\right)}{\left(r^{2}-2r-3\right)\left(r^{2}+4r\right)} by the reciprocal of \frac{r+1}{r+4}.

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

Factor the expressions that are not already factored.

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

Cancel out r\left(r+4\right)\left(r+1\right)^{2} in both numerator and denominator.

$$\frac{r-1}{r-3}$$