Multiply (3x + 5y) and (5x - 7y)

Answer

To multiply the algebraic expressions \( (3x + 5y) \) and \( (5x - 7y) \), we will use the distributive property (also known as the FOIL method for binomials). Here’s how to do it step by step:

### Step-by-Step Solution:

1. **Write the Expression**:
We start with the expression:
\[ (3x + 5y)(5x - 7y) \]

2. **Distribute Each Term**:
We will distribute each term in the first bracket with each term in the second bracket.
- First, multiply \( 3x \) by both terms in the second bracket:
\[ 3x \cdot 5x = 15x^2 \]
\[ 3x \cdot (-7y) = -21xy \]
- Next, multiply \( 5y \) by both terms in the second bracket:
\[ 5y \cdot 5x = 25xy \]
\[ 5y \cdot (-7y) = -35y^2 \]

3. **Combine All the Products**:
Now we combine all the results from the multiplications:
\[ 15x^2 - 21xy + 25xy - 35y^2 \]

4. **Combine Like Terms**:
We can now combine the like terms, specifically the \( xy \) terms:
\[ -21xy + 25xy = 4xy \]
So, we rewrite the expression:
\[ 15x^2 + 4xy - 35y^2 \]

5. **Final Result**:
The final result of multiplying \( (3x + 5y) \) and \( (5x - 7y) \) is:
\[ 15x^2 + 4xy - 35y^2 \]

### Summary:
The product of \( (3x + 5y) \) and \( (5x - 7y) \) is:
\[ 15x^2 + 4xy - 35y^2 \]