Solve 1/t^2+3t+6-5/t^2+3t+2+3/t^2+4t+3

$$\frac { 1 } { t ^ { 2 } + 3 t + 6 } - \frac { 5 } { t ^ { 2 } + 3 t + 2 } + \frac { 3 } { t ^ { 2 } + 4 t + 3 }$$

Evaluate

$$-\frac{t^{3}+9t^{2}+28t+48}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}$$

Short Solution Steps

Factor t^{2}+3t+2.

$$\frac{1}{t^{2}+3t+6}-\frac{5}{\left(t+1\right)\left(t+2\right)}+\frac{3}{t^{2}+4t+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of t^{2}+3t+6 and \left(t+1\right)\left(t+2\right) is \left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right). Multiply \frac{1}{t^{2}+3t+6} times \frac{\left(t+1\right)\left(t+2\right)}{\left(t+1\right)\left(t+2\right)}. Multiply \frac{5}{\left(t+1\right)\left(t+2\right)} times \frac{t^{2}+3t+6}{t^{2}+3t+6}.

$$\frac{\left(t+1\right)\left(t+2\right)}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}-\frac{5\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}+\frac{3}{t^{2}+4t+3}$$

Since \frac{\left(t+1\right)\left(t+2\right)}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)} and \frac{5\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(t+1\right)\left(t+2\right)-5\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}+\frac{3}{t^{2}+4t+3}$$

Do the multiplications in \left(t+1\right)\left(t+2\right)-5\left(t^{2}+3t+6\right).

$$\frac{t^{2}+2t+t+2-5t^{2}-15t-30}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}+\frac{3}{t^{2}+4t+3}$$

Combine like terms in t^{2}+2t+t+2-5t^{2}-15t-30.

$$\frac{-4t^{2}-12t-28}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}+\frac{3}{t^{2}+4t+3}$$

Factor t^{2}+4t+3.

$$\frac{-4t^{2}-12t-28}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)}+\frac{3}{\left(t+1\right)\left(t+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right) and \left(t+1\right)\left(t+3\right) is \left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right). Multiply \frac{-4t^{2}-12t-28}{\left(t+1\right)\left(t+2\right)\left(t^{2}+3t+6\right)} times \frac{t+3}{t+3}. Multiply \frac{3}{\left(t+1\right)\left(t+3\right)} times \frac{\left(t+2\right)\left(t^{2}+3t+6\right)}{\left(t+2\right)\left(t^{2}+3t+6\right)}.

$$\frac{\left(-4t^{2}-12t-28\right)\left(t+3\right)}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}+\frac{3\left(t+2\right)\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}$$

Since \frac{\left(-4t^{2}-12t-28\right)\left(t+3\right)}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)} and \frac{3\left(t+2\right)\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(-4t^{2}-12t-28\right)\left(t+3\right)+3\left(t+2\right)\left(t^{2}+3t+6\right)}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}$$

Do the multiplications in \left(-4t^{2}-12t-28\right)\left(t+3\right)+3\left(t+2\right)\left(t^{2}+3t+6\right).

$$\frac{-4t^{3}-12t^{2}-12t^{2}-36t-28t-84+3t^{3}+9t^{2}+18t+6t^{2}+18t+36}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}$$

Combine like terms in -4t^{3}-12t^{2}-12t^{2}-36t-28t-84+3t^{3}+9t^{2}+18t+6t^{2}+18t+36.

$$\frac{-t^{3}-9t^{2}-28t-48}{\left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right)}$$

Expand \left(t+1\right)\left(t+2\right)\left(t+3\right)\left(t^{2}+3t+6\right).

$$\frac{-t^{3}-9t^{2}-28t-48}{t^{5}+9t^{4}+35t^{3}+75t^{2}+84t+36}$$