1. **State the problem:** Solve the rational equation $$\frac{3m}{m+1} - \frac{5}{m+4} = \frac{2m^2 + 11m}{m^2 + 5m + 4}$$ 2. **Factor the denominator on the right side:** $$m^2 + 5m + 4 = (m+1)(m+4)$$ 3. **Rewrite the equation using the factored denominator:** $$\frac{3m}{m+1} - \frac{5}{m+4} = \frac{2m^2 + 11m}{(m+1)(m+4)}$$ 4. **Find the least common denominator (LCD):** $$\text{LCD} = (m+1)(m+4)$$ 5. **Multiply both sides of the equation by the LCD to clear denominators:** $$\cancel{(m+1)}\cancel{(m+4)} \times \left(\frac{3m}{m+1} - \frac{5}{m+4}\right) = \cancel{(m+1)}\cancel{(m+4)} \times \frac{2m^2 + 11m}{(m+1)(m+4)}$$ This simplifies to: $$3m(m+4) - 5(m+1) = 2m^2 + 11m$$ 6. **Expand the terms:** $$3m^2 + 12m - 5m - 5 = 2m^2 + 11m$$ Simplify the left side: $$3m^2 + 7m - 5 = 2m^2 + 11m$$ 7. **Bring all terms to one side:** $$3m^2 + 7m - 5 - 2m^2 - 11m = 0$$ Simplify: $$m^2 - 4m - 5 = 0$$ 8. **Factor the quadratic:** $$ (m - 5)(m + 1) = 0 $$ 9. **Solve for $m$:** $$m - 5 = 0 \Rightarrow m = 5$$ $$m + 1 = 0 \Rightarrow m = -1$$ 10. **Check for restrictions:** Denominators cannot be zero: $$m + 1 \neq 0 \Rightarrow m \neq -1$$ $$m + 4 \neq 0 \Rightarrow m \neq -4$$ Since $m = -1$ makes denominator zero, exclude it. 11. **Final solution set:** $$\boxed{\{5\}}$$