1. **State the problem:** Simplify the expression $$\frac{8}{4x + 3} + \frac{5}{6x + 1}$$ leaving denominators factorized when possible. 2. **Find the common denominator:** The denominators are $4x + 3$ and $6x + 1$. Since they cannot be factored further, the common denominator is their product: $$ (4x + 3)(6x + 1) $$ 3. **Rewrite each fraction with the common denominator:** $$ \frac{8}{4x + 3} = \frac{8(6x + 1)}{(4x + 3)(6x + 1)} $$ $$ \frac{5}{6x + 1} = \frac{5(4x + 3)}{(6x + 1)(4x + 3)} $$ 4. **Add the numerators:** $$ \frac{8(6x + 1) + 5(4x + 3)}{(4x + 3)(6x + 1)} $$ 5. **Expand the numerators:** $$ 8(6x + 1) = 48x + 8 $$ $$ 5(4x + 3) = 20x + 15 $$ 6. **Sum the expanded numerators:** $$ 48x + 8 + 20x + 15 = (48x + 20x) + (8 + 15) = 68x + 23 $$ 7. **Final simplified expression:** $$ \frac{68x + 23}{(4x + 3)(6x + 1)} $$ This is the simplified form with the denominator factorized as requested.