Joseph, let's knock this out. To solve the quadratic equation $x^2 - x - 6 = 0$ by factoring: Step 1: Identify two numbers that multiply to the constant term ($-6$) and add up to the coefficient of the $x$ term ($-1$). The two numbers are $2$ and $-3$, because $2 \times (-3) = -6$ and $2 + (-3) = -1$. Step 2: Rewrite the middle term using these two numbers. $$ x^2 + 2x - 3x - 6 = 0 $$ Step 3: Factor by grouping. Group the first two terms and the last two terms. $$ (x^2 + 2x) - (3x + 6) = 0 $$ Factor out the common term from each group. $$ x(x + 2) - 3(x + 2) = 0 $$ Step 4: Factor out the common binomial $(x+2)$. $$ (x - 3)(x + 2) = 0 $$ Step 5: Set each factor equal to zero and solve for $x$. $$ x - 3 = 0 \quad \text{or} \quad x + 2 = 0 $$ $$ x = 3 \quad \text{or} \quad x = -2 $$ The solutions are $x = 3$ and $x = -2$. The final answers are $\boxed{x = 3, x = -2}$. What's next?