To simplify the expression $\frac{5}{6x + 1} + \frac{6}{2x + 3}$, we need to find a common denominator and combine the fractions.

Steps:

1. Find the common denominator: The denominators are $6x + 1$ and $2x + 3$. The common denominator will be their product: $\text{Common denominator} = (6x + 1)(2x + 3)$

2. Rewrite each fraction with the common denominator:

   • For $\frac{5}{6x + 1}$, multiply both numerator and denominator by $2x + 3$: $\frac{5}{6x + 1} = \frac{5(2x + 3)}{(6x + 1)(2x + 3)} = \frac{10x + 15}{(6x + 1)(2x + 3)}$
   • For $\frac{6}{2x + 3}$, multiply both numerator and denominator by $6x + 1$: $\frac{6}{2x + 3} = \frac{6(6x + 1)}{(6x + 1)(2x + 3)} = \frac{36x + 6}{(6x + 1)(2x + 3)}$

3. Combine the two fractions: $\frac{10x + 15}{(6x + 1)(2x + 3)} + \frac{36x + 6}{(6x + 1)(2x + 3)} = \frac{(10x + 15) + (36x + 6)}{(6x + 1)(2x + 3)}$

4. Simplify the numerator: $(10x + 15) + (36x + 6) = 46x + 21$

5. Final simplified expression: $\frac{46x + 21}{(6x + 1)(2x + 3)}$

This is the simplified form of the given expression, with the denominator left in factored form as required.

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Related Questions:

1. How do you factor quadratic expressions?
2. How do you simplify complex rational expressions?
3. How do you solve equations involving fractions?
4. What are the properties of rational functions?
5. How can you determine the domain of a rational expression?

Tip:

When simplifying fractions, always check if the numerator can be factored further for possible cancellations with the denominator.