Solve frac{25s^{2}+10st+t^{2}}{4s^{2}-15st-4t^{2}}*frac{2s^{2}-9st+4t^{2}}{t^{2}+4st-5s^{2}}div10s^2-3st-t^2/4s^2+5st+t^2

$$\frac { 25 s ^ { 2 } + 10 s t + t ^ { 2 } } { 4 s ^ { 2 } - 15 s t - 4 t ^ { 2 } } \cdot \frac { 2 s ^ { 2 } - 9 s t + 4 t ^ { 2 } } { t ^ { 2 } + 4 s t - 5 s ^ { 2 } } \div \frac { 10 s ^ { 2 } - 3 s t - t ^ { 2 } } { 4 s ^ { 2 } + 5 s t + t ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Multiply \frac{25s^{2}+10st+t^{2}}{4s^{2}-15st-4t^{2}} times \frac{2s^{2}-9st+4t^{2}}{t^{2}+4st-5s^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)}}{\frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}}}$$

Divide \frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)} by \frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}} by multiplying \frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)} by the reciprocal of \frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}}.

$$\frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)\left(4s^{2}+5st+t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)\left(10s^{2}-3st-t^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(s+t\right)\left(s-4t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2}}{\left(s-4t\right)\left(-s+t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2}}$$

Cancel out \left(s-4t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2} in both numerator and denominator.

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

Expand

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

Short Solution Steps

Multiply \frac{25s^{2}+10st+t^{2}}{4s^{2}-15st-4t^{2}} times \frac{2s^{2}-9st+4t^{2}}{t^{2}+4st-5s^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)}}{\frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}}}$$

Divide \frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)} by \frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}} by multiplying \frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)} by the reciprocal of \frac{10s^{2}-3st-t^{2}}{4s^{2}+5st+t^{2}}.

$$\frac{\left(25s^{2}+10st+t^{2}\right)\left(2s^{2}-9st+4t^{2}\right)\left(4s^{2}+5st+t^{2}\right)}{\left(4s^{2}-15st-4t^{2}\right)\left(t^{2}+4st-5s^{2}\right)\left(10s^{2}-3st-t^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(s+t\right)\left(s-4t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2}}{\left(s-4t\right)\left(-s+t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2}}$$

Cancel out \left(s-4t\right)\left(4s+t\right)\left(2s-t\right)\left(5s+t\right)^{2} in both numerator and denominator.

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