1. **State the problem:** We want to expand and simplify the expression $$(a + \sqrt{8})^2$$ and write it in the form $$c + d\sqrt{2}$$ where $a$, $c$, and $d$ are integers. 2. **Recall the formula:** The square of a binomial is given by $$ (x + y)^2 = x^2 + 2xy + y^2 $$ 3. **Apply the formula:** Let $x = a$ and $y = \sqrt{8}$. Then $$ (a + \sqrt{8})^2 = a^2 + 2a\sqrt{8} + (\sqrt{8})^2 $$ 4. **Simplify each term:** - $a^2$ stays as is. - $2a\sqrt{8}$ can be rewritten using $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$, so $$ 2a\sqrt{8} = 2a \times 2\sqrt{2} = 4a\sqrt{2} $$ - $(\sqrt{8})^2 = 8$ 5. **Combine all terms:** $$ a^2 + 4a\sqrt{2} + 8 $$ 6. **Identify $c$ and $d$:** - The integer part without the radical is $$c = a^2 + 8$$ - The coefficient of $\sqrt{2}$ is $$d = 4a$$ **Final answer:** $$ c = a^2 + 8, \quad d = 4a $$