Solve a+3/2a+6+a+1/a^2+2a-3

$$\frac { a + 3 } { 2 a + 6 } + \frac { a + 1 } { a ^ { 2 } + 2 a - 3 }$$

Evaluate

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

Short Solution Steps

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

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

Cancel out a+3 in both numerator and denominator.

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

Factor a^{2}+2a-3.

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

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

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

Since \frac{\left(a-1\right)\left(a+3\right)}{2\left(a-1\right)\left(a+3\right)} and \frac{2\left(a+1\right)}{2\left(a-1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a-1\right)\left(a+3\right)+2\left(a+1\right).

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

Combine like terms in a^{2}+3a-a-3+2a+2.

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

Expand 2\left(a-1\right)\left(a+3\right).

$$\frac{a^{2}+4a-1}{2a^{2}+4a-6}$$

Expand

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

Short Solution Steps

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

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

Cancel out a+3 in both numerator and denominator.

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

Factor a^{2}+2a-3.

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

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

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

Since \frac{\left(a-1\right)\left(a+3\right)}{2\left(a-1\right)\left(a+3\right)} and \frac{2\left(a+1\right)}{2\left(a-1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a-1\right)\left(a+3\right)+2\left(a+1\right).

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

Combine like terms in a^{2}+3a-a-3+2a+2.

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

Expand 2\left(a-1\right)\left(a+3\right).

$$\frac{a^{2}+4a-1}{2a^{2}+4a-6}$$