1. **State the problem:** Find the points of intersection between the functions $$f(x) = x^2 + 4x$$ and $$g(x) = \sqrt{x^2 + 4x}$$. 2. **Set the functions equal to find intersection points:** $$x^2 + 4x = \sqrt{x^2 + 4x}$$ 3. **Square both sides to eliminate the square root:** $$\left(x^2 + 4x\right)^2 = x^2 + 4x$$ 4. **Expand the left side:** $$\left(x^2 + 4x\right)^2 = (x^2)^2 + 2 \cdot x^2 \cdot 4x + (4x)^2 = x^4 + 8x^3 + 16x^2$$ 5. **Rewrite the equation:** $$x^4 + 8x^3 + 16x^2 = x^2 + 4x$$ 6. **Bring all terms to one side:** $$x^4 + 8x^3 + 16x^2 - x^2 - 4x = 0$$ 7. **Simplify:** $$x^4 + 8x^3 + 15x^2 - 4x = 0$$ 8. **Factor out common factor $x$:** $$x(x^3 + 8x^2 + 15x - 4) = 0$$ 9. **Set each factor equal to zero:** - $$x = 0$$ - $$x^3 + 8x^2 + 15x - 4 = 0$$ 10. **Check $x=0$ in original functions:** $$f(0) = 0^2 + 4 \cdot 0 = 0$$ $$g(0) = \sqrt{0^2 + 4 \cdot 0} = 0$$ So, point $(0,0)$ is an intersection. 11. **Solve cubic equation:** $$x^3 + 8x^2 + 15x - 4 = 0$$ Try rational roots using factors of constant term 4: $\pm1, \pm2, \pm4$. 12. **Test $x= -1$:** $$(-1)^3 + 8(-1)^2 + 15(-1) - 4 = -1 + 8 - 15 - 4 = -12 \neq 0$$ 13. **Test $x= 1$:** $$1 + 8 + 15 - 4 = 20 \neq 0$$ 14. **Test $x= -4$:** $$(-4)^3 + 8(-4)^2 + 15(-4) - 4 = -64 + 128 - 60 - 4 = 0$$ So, $x = -4$ is a root. 15. **Divide cubic by $(x + 4)$:** Using polynomial division or synthetic division: $$x^3 + 8x^2 + 15x - 4 = (x + 4)(x^2 + 4x - 1)$$ 16. **Solve quadratic $x^2 + 4x - 1 = 0$ using quadratic formula:** $$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 + 4}}{2} = \frac{-4 \pm \sqrt{20}}{2}$$ 17. **Simplify square root:** $$\sqrt{20} = 2\sqrt{5}$$ 18. **Final roots:** $$x = \frac{-4 \pm 2\sqrt{5}}{2} = -2 \pm \sqrt{5}$$ 19. **Check these roots in original functions to ensure no extraneous solutions:** Since $g(x) = \sqrt{x^2 + 4x}$ requires $x^2 + 4x \geq 0$. Check domain for each root: - For $x = -4$: $$(-4)^2 + 4(-4) = 16 - 16 = 0 \geq 0$$ valid. - For $x = -2 + \sqrt{5}$: Calculate $x^2 + 4x$: $$(-2 + \sqrt{5})^2 + 4(-2 + \sqrt{5}) = (4 - 4\sqrt{5} + 5) - 8 + 4\sqrt{5} = 9 - 8 + 0 = 1 \geq 0$$ valid. - For $x = -2 - \sqrt{5}$: $$(-2 - \sqrt{5})^2 + 4(-2 - \sqrt{5}) = (4 + 4\sqrt{5} + 5) - 8 - 4\sqrt{5} = 9 - 8 + 0 = 1 \geq 0$$ valid. 20. **Calculate corresponding $y$ values:** Since $y = f(x) = x^2 + 4x$: - For $x = -4$: $$y = 0$$ - For $x = -2 + \sqrt{5}$: $$y = 1$$ - For $x = -2 - \sqrt{5}$: $$y = 1$$ 21. **Final intersection points:** $$(0,0), (-4,0), (-2 + \sqrt{5}, 1), (-2 - \sqrt{5}, 1)$$