solve quadratic equation by factoring

1. Identify the equation type: The given equation $x^2 + 9x + 18 = 0$ is a quadratic equation in the standard form $ax^2 + bx + c = 0$, where $a=1$, $b=9$, and $c=18$.

1. Factor the quadratic expression: To solve by factoring, we look for two numbers that multiply to $c=18$ and add up to $b=9$. The factors of $18$ are $(1, 18), (2, 9),$ and $(3, 6)$. Among these, $3 + 6 = 9$. Thus, the equation can be written as $(x + 3)(x + 6) = 0$.

3. Apply the Zero Product Property: Set each factor equal to zero to solve for $x$:
$x + 3 = 0 \implies x = -3$
$x + 6 = 0 \implies x = -6$

4. Check the solutions:
For $x = -3$: $(-3)^2 + 9(-3) + 18 = 9 - 27 + 18 = 0$. Correct.
For $x = -6$: $(-6)^2 + 9(-6) + 18 = 36 - 54 + 18 = 0$. Correct.

1. Order the solutions: The problem asks for the solutions from least to greatest. Comparing $-6$ and $-3$, we see that $-6 < -3$.