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

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Simplify the expression by following steps:
- step0: Distribute $$\sqrt{5y}$$:
  $$\left(-5a-3b\right)\sqrt{5y}$$
- step1: Distribute $$\sqrt{5y}:$$
  $$-5a\sqrt{5y}-3b\sqrt{5y}$$
Expand the expression $$ (5 a+3 b) \sqrt{5 y} $$
Simplify the expression by following steps:
- step0: Distribute $$\sqrt{5y}$$:
  $$\left(5a+3b\right)\sqrt{5y}$$
- step1: Distribute $$\sqrt{5y}:$$
  $$5a\sqrt{5y}+3b\sqrt{5y}$$
Expand the expression $$ a \sqrt{125 y}-b \sqrt{45 y} $$
Simplify the expression by following steps:
- step0: Simplify:
  $$a\sqrt{125y}-b\sqrt{45y}$$
- step1: Simplify the root:
  $$a\times 5\sqrt{5y}-b\sqrt{45y}$$
- step2: Simplify the root:
  $$a\times 5\sqrt{5y}-b\times 3\sqrt{5y}$$
- step3: Reorder the terms:
  $$5a\sqrt{5y}-b\times 3\sqrt{5y}$$
- step4: Reorder the terms:
  $$5a\sqrt{5y}-3b\sqrt{5y}$$
Expand the expression $$ (5 a-3 b) \sqrt{5 y} $$
Simplify the expression by following steps:
- step0: Distribute $$\sqrt{5y}$$:
  $$\left(5a-3b\right)\sqrt{5y}$$
- step1: Distribute $$\sqrt{5y}:$$
  $$5a\sqrt{5y}-3b\sqrt{5y}$$
Let's combine the given expressions step by step.

1. **First Expression:**
   $$
   a \sqrt{125 y} - b \sqrt{45 y}
   $$
   We can simplify this:
   $$
   \sqrt{125 y} = \sqrt{25 \cdot 5 \cdot y} = 5 \sqrt{5y}
   $$
   $$
   \sqrt{45 y} = \sqrt{9 \cdot 5 \cdot y} = 3 \sqrt{5y}
   $$
   Therefore, the first expression simplifies to:
   $$
   5a \sqrt{5y} - 3b \sqrt{5y}
   $$

2. **Second Expression:**
   $$
   (5a + 3b) \sqrt{5y}
   $$
   This expression is already in a simplified form.

3. **Third Expression:**
   $$
   (5a - 3b) \sqrt{5y}
   $$
   This expression is also in a simplified form.

4. **Fourth Expression:**
   $$
   (-5a - 3b) \sqrt{5y}
   $$
   This expression is also in a simplified form.

Now, let's combine all the simplified expressions:
- From the first expression: $$ 5a \sqrt{5y} - 3b \sqrt{5y} $$
- From the second expression: $$ 5a \sqrt{5y} + 3b \sqrt{5y} $$
- From the third expression: $$ 5a \sqrt{5y} - 3b \sqrt{5y} $$
- From the fourth expression: $$ -5a \sqrt{5y} - 3b \sqrt{5y} $$

Now, we can combine these:
$$
(5a \sqrt{5y} - 3b \sqrt{5y}) + (5a \sqrt{5y} + 3b \sqrt{5y}) + (5a \sqrt{5y} - 3b \sqrt{5y}) + (-5a \sqrt{5y} - 3b \sqrt{5y})
$$

Combining like terms:
- The $$5a \sqrt{5y}$$ terms: $$5a + 5a + 5a - 5a = 10a$$
- The $$-3b \sqrt{5y}$$ and $$+3b \sqrt{5y}$$ terms cancel out.
- The $$-3b \sqrt{5y}$$ from the first and third expressions and $$-3b \sqrt{5y}$$ from the fourth expression gives us: $$-3b - 3b = -6b$$

Thus, the final combined expression is:
$$
(10a - 6b) \sqrt{5y}
$$
The combined expression is $$ (10a - 6b) \sqrt{5y} $$.

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

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