Solve 2/q+a^2+a/a^2+4a+3+4a+3/a^2+3a

$$\frac { 2 } { q } + \frac { a ^ { 2 } + a } { a ^ { 2 } + 4 a + 3 } + \frac { 4 a + 3 } { a ^ { 2 } + 3 a }$$

Evaluate

$$\frac{aq+q+2a}{aq}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}+a}{a^{2}+4a+3}.

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

Cancel out a+1 in both numerator and denominator.

$$\frac{2}{q}+\frac{a}{a+3}+\frac{4a+3}{a^{2}+3a}$$

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

$$\frac{2\left(a+3\right)}{q\left(a+3\right)}+\frac{aq}{q\left(a+3\right)}+\frac{4a+3}{a^{2}+3a}$$

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

$$\frac{2\left(a+3\right)+aq}{q\left(a+3\right)}+\frac{4a+3}{a^{2}+3a}$$

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

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

Factor a^{2}+3a.

$$\frac{2a+6+aq}{q\left(a+3\right)}+\frac{4a+3}{a\left(a+3\right)}$$

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

$$\frac{\left(2a+6+aq\right)a}{aq\left(a+3\right)}+\frac{\left(4a+3\right)q}{aq\left(a+3\right)}$$

Since \frac{\left(2a+6+aq\right)a}{aq\left(a+3\right)} and \frac{\left(4a+3\right)q}{aq\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2a+6+aq\right)a+\left(4a+3\right)q}{aq\left(a+3\right)}$$

Do the multiplications in \left(2a+6+aq\right)a+\left(4a+3\right)q.

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

Factor the expressions that are not already factored in \frac{2a^{2}+6a+a^{2}q+4aq+3q}{aq\left(a+3\right)}.

$$\frac{\left(a+3\right)\left(aq+q+2a\right)}{aq\left(a+3\right)}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{aq+q+2a}{aq}$$

Expand

$$\frac{aq+q+2a}{aq}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}+a}{a^{2}+4a+3}.

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

Cancel out a+1 in both numerator and denominator.

$$\frac{2}{q}+\frac{a}{a+3}+\frac{4a+3}{a^{2}+3a}$$

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

$$\frac{2\left(a+3\right)}{q\left(a+3\right)}+\frac{aq}{q\left(a+3\right)}+\frac{4a+3}{a^{2}+3a}$$

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

$$\frac{2\left(a+3\right)+aq}{q\left(a+3\right)}+\frac{4a+3}{a^{2}+3a}$$

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

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

Factor a^{2}+3a.

$$\frac{2a+6+aq}{q\left(a+3\right)}+\frac{4a+3}{a\left(a+3\right)}$$

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

$$\frac{\left(2a+6+aq\right)a}{aq\left(a+3\right)}+\frac{\left(4a+3\right)q}{aq\left(a+3\right)}$$

Since \frac{\left(2a+6+aq\right)a}{aq\left(a+3\right)} and \frac{\left(4a+3\right)q}{aq\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2a+6+aq\right)a+\left(4a+3\right)q}{aq\left(a+3\right)}$$

Do the multiplications in \left(2a+6+aq\right)a+\left(4a+3\right)q.

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

Factor the expressions that are not already factored in \frac{2a^{2}+6a+a^{2}q+4aq+3q}{aq\left(a+3\right)}.

$$\frac{\left(a+3\right)\left(aq+q+2a\right)}{aq\left(a+3\right)}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{aq+q+2a}{aq}$$