Question: User: $$x^2 + 5x + 6 = 0$$ $$(x + 2)(x + 3) = 0 \implies x = -2, \; x = -3$$ $$\text{Domain: } \mathbb{R} \setminus \{-3, -2\}$$
1. **State the problem:** Solve the quadratic equation $$x^2 + 5x + 6 = 0$$ and find its roots.
2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can factor it if possible or use the quadratic formula. Here, factoring is possible.
3. **Factor the quadratic:**
$$x^2 + 5x + 6 = (x + 2)(x + 3)$$
4. **Set each factor equal to zero:**
$$(x + 2) = 0 \implies x = -2$$
$$(x + 3) = 0 \implies x = -3$$
5. **Interpret the solution:** The roots of the equation are $$x = -2$$ and $$x = -3$$.
6. **Domain note:** The domain given is $$\mathbb{R} \setminus \{-3, -2\}$$, which means all real numbers except $$-3$$ and $$-2$$. This is typically used when these values cause undefined expressions in a function, but for this quadratic equation, the roots are simply the solutions.
**Final answer:** $$x = -2, \quad x = -3$$