1. **State the problem:** We need to translate the equation "square root of (x + 4) + square root of (x + 2) = 6" into a mathematical equation and then solve for $x$. 2. **Write the equation:** The square root of a number $a$ is written as $\sqrt{a}$. So the equation becomes: $$\sqrt{x+4} + \sqrt{x+2} = 6$$ 3. **Isolate one square root:** To solve, isolate one of the square roots. Let's isolate $\sqrt{x+4}$: $$\sqrt{x+4} = 6 - \sqrt{x+2}$$ 4. **Square both sides:** Squaring both sides removes the square root on the left: $$\left(\sqrt{x+4}\right)^2 = \left(6 - \sqrt{x+2}\right)^2$$ $$x + 4 = 36 - 12\sqrt{x+2} + (x + 2)$$ 5. **Simplify:** Combine like terms: $$x + 4 = 36 + x + 2 - 12\sqrt{x+2}$$ $$x + 4 = x + 38 - 12\sqrt{x+2}$$ 6. **Subtract $x$ from both sides:** $$4 = 38 - 12\sqrt{x+2}$$ 7. **Isolate the square root term:** $$-34 = -12\sqrt{x+2}$$ $$\sqrt{x+2} = \frac{34}{12} = \frac{17}{6}$$ 8. **Square both sides again:** $$x + 2 = \left(\frac{17}{6}\right)^2 = \frac{289}{36}$$ 9. **Solve for $x$:** $$x = \frac{289}{36} - 2 = \frac{289}{36} - \frac{72}{36} = \frac{217}{36}$$ 10. **Check the solution:** Substitute $x = \frac{217}{36}$ back into the original equation: $$\sqrt{\frac{217}{36} + 4} + \sqrt{\frac{217}{36} + 2} = \sqrt{\frac{217}{36} + \frac{144}{36}} + \sqrt{\frac{217}{36} + \frac{72}{36}} = \sqrt{\frac{361}{36}} + \sqrt{\frac{289}{36}} = \frac{19}{6} + \frac{17}{6} = 6$$ The solution satisfies the original equation. **Final answer:** $$x = \frac{217}{36}$$