1. **State the problem:** Solve the quadratic equation $2a^2 = 6 + 8a$ by completing the square. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2a^2 - 8a - 6 = 0$$ 3. **Divide through by the coefficient of $a^2$ to simplify:** $$\cancel{2}a^2 - \cancel{2} \times 4a - \cancel{2} \times 3 = 0 \implies a^2 - 4a - 3 = 0$$ 4. **Isolate the constant term:** $$a^2 - 4a = 3$$ 5. **Complete the square:** Take half of the coefficient of $a$, which is $-4$, half is $-2$, then square it: $(-2)^2 = 4$. Add 4 to both sides: $$a^2 - 4a + 4 = 3 + 4$$ 6. **Rewrite the left side as a perfect square:** $$(a - 2)^2 = 7$$ 7. **Take the square root of both sides:** $$a - 2 = \pm \sqrt{7}$$ 8. **Solve for $a$:** $$a = 2 \pm \sqrt{7}$$ **Final answer:** $$a = 2 + \sqrt{7} \quad \text{or} \quad a = 2 - \sqrt{7}$$