1. **Problem:** Simplify the algebraic fraction $\frac{3x^2 + 7x + 4}{x^2 + 3x + 2}$. 2. **Formula and rules:** To simplify algebraic fractions, factor numerator and denominator completely and cancel common factors. 3. **Factor numerator:** $$3x^2 + 7x + 4 = (3x + 4)(x + 1)$$ 4. **Factor denominator:** $$x^2 + 3x + 2 = (x + 1)(x + 2)$$ 5. **Rewrite fraction:** $$\frac{(3x + 4)(x + 1)}{(x + 1)(x + 2)}$$ 6. **Cancel common factor $x + 1$:** $$\frac{\cancel{(x + 1)}(3x + 4)}{\cancel{(x + 1)}(x + 2)} = \frac{3x + 4}{x + 2}$$ 7. **Final answer:** $$\boxed{\frac{3x + 4}{x + 2}}$$ This is the simplified form of the given algebraic fraction.