1. **Problem Statement:** Solve the equation $$(x + 3)(x + 5)(x + 7)(x + 9) = 9.$$ 2. **Step 1: Introduce substitutions for simplification.** Notice the expression involves four consecutive even-spaced terms. Let's set $$y = x + 6$$ so the factors become $$(y - 3)(y - 1)(y + 1)(y + 3).$$ 3. **Step 2: Use difference of squares to simplify pairs.** Group the terms as follows: $$(y - 3)(y + 3) = y^2 - 9$$ $$(y - 1)(y + 1) = y^2 - 1$$ So the left side becomes: $$ (y^2 - 9)(y^2 - 1).$$ 4. **Step 3: Multiply the two quadratic expressions.** Using the distributive property (FOIL), we get: $$y^4 - y^2 - 9y^2 + 9 = y^4 - 10y^2 + 9.$$ 5. **Step 4: Set the equation equal to 9 and simplify:** $$y^4 - 10y^2 + 9 = 9 \\ y^4 - 10y^2 + 9 - 9 = 0 \\ y^4 - 10y^2 = 0.$$ 6. **Step 5: Factor the equation:** $$y^2 (y^2 - 10) = 0.$$ 7. **Step 6: Solve for $y$: ** - $y^2 = 0$ gives $y=0$ - $y^2 -10=0$ gives $y^2=10$ so $y = \pm \sqrt{10}$ 8. **Step 7: Recall substitution to solve for $x$:** $$x + 6 = 0 \implies x = -6,$$ $$x + 6 = \sqrt{10} \implies x = -6 + \sqrt{10},$$ $$x + 6 = -\sqrt{10} \implies x = -6 - \sqrt{10}.$$ **Final solutions:** $$x = -6, \quad x = -6 + \sqrt{10}, \quad x = -6 - \sqrt{10}.$$