1. **State the problem:** Simplify the expression $(5 + \sqrt{12})(11 + \sqrt{3})$ and write it in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. 2. **Recall the formula:** Use the distributive property (FOIL) to expand the product: $$ (x + y)(m + n) = xm + xn + ym + yn $$ 3. **Apply the formula:** $$ (5 + \sqrt{12})(11 + \sqrt{3}) = 5 \times 11 + 5 \times \sqrt{3} + \sqrt{12} \times 11 + \sqrt{12} \times \sqrt{3} $$ 4. **Calculate each term:** - $5 \times 11 = 55$ - $5 \times \sqrt{3} = 5\sqrt{3}$ - $\sqrt{12} \times 11 = 11\sqrt{12}$ - $\sqrt{12} \times \sqrt{3} = \sqrt{12 \times 3} = \sqrt{36} = 6$ 5. **Simplify $\sqrt{12}$:** $$ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} $$ 6. **Substitute back:** $$ 55 + 5\sqrt{3} + 11 \times 2\sqrt{3} + 6 = 55 + 5\sqrt{3} + 22\sqrt{3} + 6 $$ 7. **Combine like terms:** - Combine constants: $55 + 6 = 61$ - Combine $\sqrt{3}$ terms: $5\sqrt{3} + 22\sqrt{3} = 27\sqrt{3}$ 8. **Final expression:** $$ 61 + 27\sqrt{3} $$ **Answer:** $a = 61$, $b = 27$