1. **State the problem:** Solve the quadratic equation $$x^2 + 8x = -4$$ for $x$. 2. **Rewrite the equation in standard form:** Move all terms to one side: $$x^2 + 8x + 4 = 0$$ 3. **Methods to solve quadratic equations:** - **Factoring:** Try to express the quadratic as a product of two binomials. - **Quadratic Formula:** Use the formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=8$, $c=4$. - **Graphing:** Rewrite as a function $$f(x) = x^2 + 8x + 4$$ and find the $x$-intercepts. - **Completing the Square:** Rewrite the quadratic in the form $$(x + d)^2 = e$$ and solve. 4. **Check if factoring works:** We look for two numbers that multiply to $4$ and add to $8$. Since $4$ and $8$ do not factor nicely to satisfy this, factoring is not straightforward here. 5. **Quadratic Formula:** $$x = \frac{-8 \pm \sqrt{8^2 - 4 \times 1 \times 4}}{2 \times 1} = \frac{-8 \pm \sqrt{64 - 16}}{2} = \frac{-8 \pm \sqrt{48}}{2}$$ Simplify $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$$: $$x = \frac{-8 \pm 4\sqrt{3}}{2}$$ Cancel common factor 2: $$x = \frac{\cancel{2}(-4 \pm 2\sqrt{3})}{\cancel{2}} = -4 \pm 2\sqrt{3}$$ 6. **Completing the Square:** Start with $$x^2 + 8x + 4 = 0$$ Move constant: $$x^2 + 8x = -4$$ Add $$\left(\frac{8}{2}\right)^2 = 16$$ to both sides: $$x^2 + 8x + 16 = -4 + 16$$ $$ (x + 4)^2 = 12$$ Take square root: $$x + 4 = \pm \sqrt{12} = \pm 2\sqrt{3}$$ Solve for $x$: $$x = -4 \pm 2\sqrt{3}$$ 7. **Graphing:** Plotting $$y = x^2 + 8x + 4$$ and finding $x$-intercepts will give the same solutions. **Answer:** The methods that work are: Using the Quadratic Formula, Rewriting as a function and graphing it, and Completing the Square Method. Factoring does not work easily here.