Solve r^2-7r+12/r^2+r-20*r^2+10r+25/24-5r-r^2

$$\frac { r ^ { 2 } - 7 r + 12 } { r ^ { 2 } + r - 20 } \cdot \frac { r ^ { 2 } + 10 r + 25 } { 24 - 5 r - r ^ { 2 } }$$

Evaluate

$$-\frac{r+5}{r+8}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{r^{2}-7r+12}{r^{2}+r-20}.

$$\frac{\left(r-4\right)\left(r-3\right)}{\left(r-4\right)\left(r+5\right)}\times \frac{r^{2}+10r+25}{24-5r-r^{2}}$$

Cancel out r-4 in both numerator and denominator.

$$\frac{r-3}{r+5}\times \frac{r^{2}+10r+25}{24-5r-r^{2}}$$

Multiply \frac{r-3}{r+5} times \frac{r^{2}+10r+25}{24-5r-r^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r-3\right)\left(r^{2}+10r+25\right)}{\left(r+5\right)\left(24-5r-r^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(r-3\right)\left(r+5\right)^{2}}{\left(r+5\right)\left(r+8\right)\left(-r+3\right)}$$

Extract the negative sign in -3+r.

$$\frac{-\left(-r+3\right)\left(r+5\right)^{2}}{\left(r+5\right)\left(r+8\right)\left(-r+3\right)}$$

Cancel out \left(r+5\right)\left(-r+3\right) in both numerator and denominator.

$$\frac{-\left(r+5\right)}{r+8}$$

Expand the expression.

$$\frac{-r-5}{r+8}$$

Expand

$$-\frac{r+5}{r+8}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{r^{2}-7r+12}{r^{2}+r-20}.

$$\frac{\left(r-4\right)\left(r-3\right)}{\left(r-4\right)\left(r+5\right)}\times \frac{r^{2}+10r+25}{24-5r-r^{2}}$$

Cancel out r-4 in both numerator and denominator.

$$\frac{r-3}{r+5}\times \frac{r^{2}+10r+25}{24-5r-r^{2}}$$

Multiply \frac{r-3}{r+5} times \frac{r^{2}+10r+25}{24-5r-r^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(r-3\right)\left(r^{2}+10r+25\right)}{\left(r+5\right)\left(24-5r-r^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(r-3\right)\left(r+5\right)^{2}}{\left(r+5\right)\left(r+8\right)\left(-r+3\right)}$$

Extract the negative sign in -3+r.

$$\frac{-\left(-r+3\right)\left(r+5\right)^{2}}{\left(r+5\right)\left(r+8\right)\left(-r+3\right)}$$

Cancel out \left(r+5\right)\left(-r+3\right) in both numerator and denominator.

$$\frac{-\left(r+5\right)}{r+8}$$

Expand the expression.

$$\frac{-r-5}{r+8}$$