Problem 67 - College Algebra

Question

Find each product. $$(x+5 y)(7 x+3 y)$$

Step-by-Step Solution

Answer

The product of \( (x+5y)(7x+3y) \) is \( 7x^2 + 38xy + 15y^2 \).

Step 1: Distributing the first term of the first binomial

First, we distribute the \( x \) of the first binomial over the second binomial. That gives us \( x(7x) = 7x^2 \) and \( x(3y) = 3xy \). This results in an intermediate product of \( 7x^2 + 3xy \).

Step 2: Distributing the second term of the first binomial

Next, distribute the \( 5y \) of the first binomial over the second binomial. That gives us \(5y(7x) = 35xy \) and \( 5y(3y) = 15y^2 \). This results in a second intermediate product of \( 35xy + 15y^2 \).

Step 3: Adding the intermediate products

We then combine the intermediate products: \( (7x^2 + 3xy) + (35xy + 15y^2) \)

Step 4: Simplify

In this step, combine like terms. When we add \( 7x^2 \), \( 3xy \), \( 35xy \), and \( 15y^2 \), we get \( 7x^2 + 38xy + 15y^2 \).