Multiply
`5a + 3b " by " 7a - 4b`

Answer

To solve the problem of multiplying the algebraic expressions \(5a + 3b\) by \(7a - 4b\), we will use the distributive property. Here are the steps:

### Step 1: Write down the expression to be multiplied
We start with the expression:
\[ (5a + 3b)(7a - 4b) \]

### Step 2: Distribute each term in the first expression to each term in the second expression
We will multiply \(5a\) by both terms in \(7a - 4b\) and then \(3b\) by both terms in \(7a - 4b\).

1. Multiply \(5a\) by \(7a\):
\[ 5a \cdot 7a = 35a^2 \]

2. Multiply \(5a\) by \(-4b\):
\[ 5a \cdot (-4b) = -20ab \]

3. Multiply \(3b\) by \(7a\):
\[ 3b \cdot 7a = 21ab \]

4. Multiply \(3b\) by \(-4b\):
\[ 3b \cdot (-4b) = -12b^2 \]

### Step 3: Combine all the results
Now we combine all the terms we calculated:
\[ 35a^2 - 20ab + 21ab - 12b^2 \]

### Step 4: Combine like terms
Next, we combine the like terms, which are \(-20ab\) and \(21ab\):
\[ -20ab + 21ab = 1ab \quad \text{or simply } ab \]

So, the expression simplifies to:
\[ 35a^2 + ab - 12b^2 \]

### Final Answer
Thus, the final result of multiplying \(5a + 3b\) by \(7a - 4b\) is:
\[ 35a^2 + ab - 12b^2 \]

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