Solve frac{t^{2}+3t}{t^{2}-t^{4}}*t^2+t^3/t^3+6t^2+9t

$$\frac { t ^ { 2 } + 3 t } { t ^ { 2 } - t ^ { 4 } } \cdot \frac { t ^ { 2 } + t ^ { 3 } } { t ^ { 3 } + 6 t ^ { 2 } + 9 t }$$

Evaluate

$$-\frac{1}{\left(t-1\right)\left(t+3\right)}$$

Short Solution Steps

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

$$\frac{t\left(t+3\right)}{\left(t-1\right)\left(-t-1\right)t^{2}}\times \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}$$

Cancel out t in both numerator and denominator.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}$$

Factor the expressions that are not already factored in \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{\left(t+1\right)t^{2}}{t\left(t+3\right)^{2}}$$

Cancel out t in both numerator and denominator.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{t\left(t+1\right)}{\left(t+3\right)^{2}}$$

Multiply \frac{t+3}{t\left(t-1\right)\left(-t-1\right)} times \frac{t\left(t+1\right)}{\left(t+3\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in t+1.

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

Cancel out t\left(-t-1\right)\left(t+3\right) in both numerator and denominator.

$$\frac{-1}{\left(t-1\right)\left(t+3\right)}$$

Use the distributive property to multiply t-1 by t+3 and combine like terms.

$$\frac{-1}{t^{2}+2t-3}$$

Expand

$$\frac{1}{\left(1-t\right)\left(t+3\right)}$$

Short Solution Steps

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

$$\frac{t\left(t+3\right)}{\left(t-1\right)\left(-t-1\right)t^{2}}\times \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}$$

Cancel out t in both numerator and denominator.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}$$

Factor the expressions that are not already factored in \frac{t^{2}+t^{3}}{t^{3}+6t^{2}+9t}.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{\left(t+1\right)t^{2}}{t\left(t+3\right)^{2}}$$

Cancel out t in both numerator and denominator.

$$\frac{t+3}{t\left(t-1\right)\left(-t-1\right)}\times \frac{t\left(t+1\right)}{\left(t+3\right)^{2}}$$

Multiply \frac{t+3}{t\left(t-1\right)\left(-t-1\right)} times \frac{t\left(t+1\right)}{\left(t+3\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Extract the negative sign in t+1.

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

Cancel out t\left(-t-1\right)\left(t+3\right) in both numerator and denominator.

$$\frac{-1}{\left(t-1\right)\left(t+3\right)}$$

Use the distributive property to multiply t-1 by t+3 and combine like terms.

$$\frac{-1}{t^{2}+2t-3}$$