Expand the Expression: (a-5b)(7a+3b) Using Binomial Multiplication

In order to solve the given problem, we'll follow these steps:

• Step 1: Use the distributive property to expand the expression.

• Step 2: Combine like terms to simplify the expression.

Let's proceed to work through each step:

Step 1: Begin by applying the distributive property to expand the expression:

$(a-5b)(7a+3b) = a(7a) + a(3b) - 5b(7a) - 5b(3b)$

Calculate each term:

• $a \cdot 7a = 7a^2$

• $a \cdot 3b = 3ab$

• $-5b \cdot 7a = -35ab$

• $-5b \cdot 3b = -15b^2$

Merge together, as follows:

$7a^2 + 3ab - 35ab - 15b^2$

Step 2: Combine like terms:

The terms $3ab$ and $-35ab$ are like terms, hence we combine them:

$7a^2 + (3ab - 35ab) - 15b^2 = 7a^2 - 32ab - 15b^2$

Therefore, the solution to the problem is $7a^2 - 32ab - 15b^2$.