Solve 49-t^2/t^2+6t+9divt^2-4t-21/t^2+3t

$$\frac { 49 - t ^ { 2 } } { t ^ { 2 } + 6 t + 9 } \div \frac { t ^ { 2 } - 4 t - 21 } { t ^ { 2 } + 3 t }$$

Evaluate

$$-\frac{t\left(t+7\right)}{\left(t+3\right)^{2}}$$

Short Solution Steps

Divide \frac{49-t^{2}}{t^{2}+6t+9} by \frac{t^{2}-4t-21}{t^{2}+3t} by multiplying \frac{49-t^{2}}{t^{2}+6t+9} by the reciprocal of \frac{t^{2}-4t-21}{t^{2}+3t}.

$$\frac{\left(49-t^{2}\right)\left(t^{2}+3t\right)}{\left(t^{2}+6t+9\right)\left(t^{2}-4t-21\right)}$$

Factor the expressions that are not already factored.

$$\frac{t\left(t-7\right)\left(-t-7\right)\left(t+3\right)}{\left(t-7\right)\left(t+3\right)\left(t+3\right)^{2}}$$

Cancel out \left(t-7\right)\left(t+3\right) in both numerator and denominator.

$$\frac{t\left(-t-7\right)}{\left(t+3\right)^{2}}$$

Expand the expression.

$$\frac{-t^{2}-7t}{t^{2}+6t+9}$$

Expand

$$-\frac{t^{2}+7t}{\left(t+3\right)^{2}}$$

Short Solution Steps

Divide \frac{49-t^{2}}{t^{2}+6t+9} by \frac{t^{2}-4t-21}{t^{2}+3t} by multiplying \frac{49-t^{2}}{t^{2}+6t+9} by the reciprocal of \frac{t^{2}-4t-21}{t^{2}+3t}.

$$\frac{\left(49-t^{2}\right)\left(t^{2}+3t\right)}{\left(t^{2}+6t+9\right)\left(t^{2}-4t-21\right)}$$

Factor the expressions that are not already factored.

$$\frac{t\left(t-7\right)\left(-t-7\right)\left(t+3\right)}{\left(t-7\right)\left(t+3\right)\left(t+3\right)^{2}}$$

Cancel out \left(t-7\right)\left(t+3\right) in both numerator and denominator.

$$\frac{t\left(-t-7\right)}{\left(t+3\right)^{2}}$$

Expand the expression.

$$\frac{-t^{2}-7t}{t^{2}+6t+9}$$