1. **State the problem:** Solve the quadratic equation $$x^2 - 22 = 5x + 2$$ by factoring. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 22 - 5x - 2 = 0$$ Simplify: $$x^2 - 5x - 24 = 0$$ 3. **Identify the quadratic form:** The equation is now in standard form: $$ax^2 + bx + c = 0$$ where $a=1$, $b=-5$, and $c=-24$. 4. **Factor the quadratic:** Find two numbers that multiply to $ac = 1 \times (-24) = -24$ and add to $b = -5$. These numbers are $-8$ and $3$ because $-8 \times 3 = -24$ and $-8 + 3 = -5$. 5. **Rewrite the middle term using these numbers:** $$x^2 - 8x + 3x - 24 = 0$$ 6. **Group terms and factor each group:** $$x(x - 8) + 3(x - 8) = 0$$ 7. **Factor out the common binomial:** $$(x - 8)(x + 3) = 0$$ 8. **Apply the zero product property:** Set each factor equal to zero: $$x - 8 = 0 \quad \Rightarrow \quad x = 8$$ $$x + 3 = 0 \quad \Rightarrow \quad x = -3$$ **Final answer:** The solutions to the equation are $$x = 8$$ and $$x = -3$$.