https://file.aiquickdraw.com/mathbot/file/17273538096177smiyh5o.jpg

The equation shown in the image is:

\[
(3p - 6q)(5p - q) - (3p - q)^2
\]

Let's solve it step by step.

### Step 1: Expand the first product \((3p - 6q)(5p - q)\)
We use the distributive property (FOIL method):

\[
(3p - 6q)(5p - q) = 3p(5p - q) - 6q(5p - q)
\]
\[
= 3p(5p) + 3p(-q) - 6q(5p) - 6q(-q)
\]
\[
= 15p^2 - 3pq - 30pq + 6q^2
\]
\[
= 15p^2 - 33pq + 6q^2
\]

### Step 2: Expand \((3p - q)^2\)
We use the formula \((a - b)^2 = a^2 - 2ab + b^2\):

\[
(3p - q)^2 = (3p)^2 - 2(3p)(q) + (q)^2
\]
\[
= 9p^2 - 6pq + q^2
\]

### Step 3: Subtract the result of the square from the first expanded expression
Now, we subtract:

\[
(15p^2 - 33pq + 6q^2) - (9p^2 - 6pq + q^2)
\]
Distribute the negative sign:

\[
= 15p^2 - 33pq + 6q^2 - 9p^2 + 6pq - q^2
\]
Now, combine like terms:

\[
= (15p^2 - 9p^2) + (-33pq + 6pq) + (6q^2 - q^2)
\]
\[
= 6p^2 - 27pq + 5q^2
\]

Thus, the simplified expression is:

\[
6p^2 - 27pq + 5q^2
\]

Would you like further clarification or details on this solution?

### Related questions:
1. How does the distributive property work for binomials?
2. What are the steps in applying the FOIL method?
3. How do you simplify expressions after expanding terms?
4. What is the formula for expanding a square of a binomial?
5. How do you combine like terms after distributing?

**Tip:** When expanding binomials, carefully apply the distributive property or the FOIL method to ensure all terms are accounted for.

Learn how to simplify the algebraic expression (3p - 6q)(5p - q) - (3p - q)^2 using the distributive property and binomial expansion. This problem involves algebraic concepts suitable for students in Grades 9-12.

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