1. **State the problem:** We are given the equation $$(e - 2\sqrt{3})^2 = f - 20\sqrt{3}$$ where $e$ and $f$ are integers. We need to find the values of $e$ and $f$. 2. **Expand the left side:** Use the formula for the square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ Here, $a = e$ and $b = 2\sqrt{3}$. So, $$ (e - 2\sqrt{3})^2 = e^2 - 2 \times e \times 2\sqrt{3} + (2\sqrt{3})^2 $$ 3. **Calculate each term:** $$ = e^2 - 4e\sqrt{3} + 4 \times 3 = e^2 - 4e\sqrt{3} + 12 $$ 4. **Rewrite the equation:** $$ e^2 - 4e\sqrt{3} + 12 = f - 20\sqrt{3} $$ 5. **Match rational and irrational parts:** Since $e$ and $f$ are integers, the rational parts and the coefficients of $\sqrt{3}$ on both sides must be equal separately. - Rational parts: $$e^2 + 12 = f$$ - Irrational parts: $$-4e = -20$$ 6. **Solve for $e$:** $$-4e = -20 \implies e = \frac{-20}{-4} = 5$$ 7. **Find $f$ using $e=5$:** $$f = e^2 + 12 = 5^2 + 12 = 25 + 12 = 37$$ **Final answer:** $$e = 5, \quad f = 37$$