Combine the following expressions. \( \begin{array}{l}a \sqrt{125 y}-b \sqrt{45 y} \\ (5 a+3 b) \sqrt{5 y} \\ (5 a-3 b) \sqrt{5 y} \\ (-5 a-3 b) \sqrt{5 y}\end{array} \)

To combine the given expressions, we start by simplifying each term involving the square roots. 1. Simplify \( a \sqrt{125y} \): \[ a \sqrt{125y} = a \sqrt{25 \cdot 5 \cdot y} = a \cdot 5 \sqrt{5y} = 5a \sqrt{5y} \] 2. Simplify \( -b \sqrt{45y} \): \[ -b \sqrt{45y} = -b \sqrt{9 \cdot 5 \cdot y} = -b \cdot 3 \sqrt{5y} = -3b \sqrt{5y} \] So, the first expression \( a \sqrt{125y} - b \sqrt{45y} \) can be combined as: \[ 5a \sqrt{5y} - 3b \sqrt{5y} = (5a - 3b) \sqrt{5y} \] Next, we combine the second, third, and fourth terms that already have the common factor \( \sqrt{5y} \): - \( (5a + 3b) \sqrt{5y} \) - \( (5a - 3b) \sqrt{5y} \) - \( (-5a - 3b) \sqrt{5y} \) Now, let's combine these: \[ (5a - 3b) \sqrt{5y} + (5a + 3b) \sqrt{5y} + (-5a - 3b) \sqrt{5y} \] We can factor out \( \sqrt{5y} \): \[ \sqrt{5y} \left((5a - 3b) + (5a + 3b) + (-5a - 3b)\right) \] Next, we combine the coefficients: \[ (5a - 3b + 5a + 3b - 5a - 3b) = (5a + 5a - 5a) + (-3b + 3b - 3b) = 5a - 3b \] Finally, we include the simplified first expression: \[ (5a - 3b) \sqrt{5y} + (5a - 3b) \sqrt{5y} = 2(5a - 3b) \sqrt{5y} \] Thus, the combined expression is: \[ 2(5a - 3b) \sqrt{5y} \]