1. **State the problem:** Simplify the expression $$\frac{2x^2 + 9x + 4}{x + 4}$$. 2. **Recall the formula:** To simplify a rational expression, factor the numerator and denominator if possible, then cancel common factors. 3. **Factor the numerator:** We look for two numbers that multiply to $2 \times 4 = 8$ and add to $9$. These are $8$ and $1$. 4. Rewrite the numerator: $$2x^2 + 9x + 4 = 2x^2 + 8x + x + 4$$ 5. Group terms: $$= (2x^2 + 8x) + (x + 4)$$ 6. Factor each group: $$= 2x(x + 4) + 1(x + 4)$$ 7. Factor out the common binomial: $$= (2x + 1)(x + 4)$$ 8. **Rewrite the original expression:** $$\frac{(2x + 1)(x + 4)}{x + 4}$$ 9. **Cancel the common factor $x + 4$:** $$= \frac{\cancel{(x + 4)}(2x + 1)}{\cancel{x + 4}} = 2x + 1$$ 10. **Final answer:** $$2x + 1$$ This simplification is valid for all $x \neq -4$ because division by zero is undefined.