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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Combine like terms in 4a^{2}+6a+a^{4}+3a^{3}-a^{3}-3a^{2}.

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

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

$$\frac{a^{2}+6a+a^{4}+2a^{3}}{2a^{3}+7a^{2}-9}$$

Factor

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