Solve 2t/t-4-5t/t^2+6

$$\frac { 2 t } { t - 4 } - \frac { 5 t } { t ^ { 2 } + 6 }$$

Evaluate

$$\frac{t\left(2t^{2}-5t+32\right)}{\left(t-4\right)\left(t^{2}+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of t-4 and t^{2}+6 is \left(t-4\right)\left(t^{2}+6\right). Multiply \frac{2t}{t-4} times \frac{t^{2}+6}{t^{2}+6}. Multiply \frac{5t}{t^{2}+6} times \frac{t-4}{t-4}.

$$\frac{2t\left(t^{2}+6\right)}{\left(t-4\right)\left(t^{2}+6\right)}-\frac{5t\left(t-4\right)}{\left(t-4\right)\left(t^{2}+6\right)}$$

Since \frac{2t\left(t^{2}+6\right)}{\left(t-4\right)\left(t^{2}+6\right)} and \frac{5t\left(t-4\right)}{\left(t-4\right)\left(t^{2}+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2t\left(t^{2}+6\right)-5t\left(t-4\right)}{\left(t-4\right)\left(t^{2}+6\right)}$$

Do the multiplications in 2t\left(t^{2}+6\right)-5t\left(t-4\right).

$$\frac{2t^{3}+12t-5t^{2}+20t}{\left(t-4\right)\left(t^{2}+6\right)}$$

Combine like terms in 2t^{3}+12t-5t^{2}+20t.

$$\frac{2t^{3}+32t-5t^{2}}{\left(t-4\right)\left(t^{2}+6\right)}$$

Expand \left(t-4\right)\left(t^{2}+6\right).

$$\frac{2t^{3}+32t-5t^{2}}{t^{3}-4t^{2}+6t-24}$$

Factor

$$\frac{t\left(2t^{2}-5t+32\right)}{\left(t-4\right)\left(t^{2}+6\right)}$$