1. **State the problem:** We are given the equation $(3 + \sqrt{a})(4 + \sqrt{a}) = 17 + k\sqrt{a}$ where $a$ and $k$ are positive integers. We need to find the values of $a$ and $k$. 2. **Expand the left side:** Use the distributive property (FOIL) to expand: $$ (3 + \sqrt{a})(4 + \sqrt{a}) = 3 \times 4 + 3 \times \sqrt{a} + 4 \times \sqrt{a} + \sqrt{a} \times \sqrt{a} $$ which simplifies to $$ 12 + 3\sqrt{a} + 4\sqrt{a} + a $$ 3. **Combine like terms:** The terms with $\sqrt{a}$ combine: $$ 12 + a + 7\sqrt{a} $$ 4. **Set equal to the right side:** The equation becomes $$ 12 + a + 7\sqrt{a} = 17 + k\sqrt{a} $$ 5. **Equate rational and irrational parts:** Since $a$ and $k$ are integers, the rational parts and the coefficients of $\sqrt{a}$ must be equal separately: - Rational parts: $12 + a = 17$ - Irrational parts: $7 = k$ 6. **Solve for $a$:** $$ a = 17 - 12 = 5 $$ 7. **Solve for $k$:** $$ k = 7 $$ **Final answer:** $$ a = 5, \quad k = 7 $$