There is a minus sign in front of $(p - q)(5p + q)$. That minus sign must be distributed to every term in the expansion of that product.

After distributing and expanding, collect like terms: group together the $p^2$ terms, the $pq$ terms, and the $q^2$ terms, and then add their coefficients.

Pay extra attention to the signs on the $q^2$ terms when you combine them; a sign mistake here is easy to make and will change the final answer.

In Desmos, choose simple test values, for example type p = 2 and on the next line q = 3. You can change these numbers later to double-check.

On a new line, type the original expression using your $p$ and $q$ values: (5p - 2q)(p + 3q) - (p - q)(5p + q) + 4q(5p - q). Note the numerical result Desmos gives.

For each answer choice, type its expression using the same $p$ and $q$ values (for example, 37p q - 9q^2 for one choice). Compare the numerical result of each choice to the result of the original expression. The correct choice will match the original expression’s value for several different pairs of $p$ and $q$ you try.

First, expand each product using the distributive property.

• For $(5p - 2q)(p + 3q)$:

  • $5p \cdot p = 5p^2$
  • $5p \cdot 3q = 15pq$
  • $-2q \cdot p = -2pq$
  • $-2q \cdot 3q = -6q^2$ So this becomes $5p^2 + 15pq - 2pq - 6q^2 = 5p^2 + 13pq - 6q^2$.

• For $(p - q)(5p + q)$:

  • $p \cdot 5p = 5p^2$
  • $p \cdot q = pq$
  • $-q \cdot 5p = -5pq$
  • $-q \cdot q = -q^2$ So this becomes $5p^2 + pq - 5pq - q^2 = 5p^2 - 4pq - q^2$.

• For $4q(5p - q)$:

  • $4q \cdot 5p = 20pq$
  • $4q \cdot (-q) = -4q^2$ So this becomes $20pq - 4q^2$.

Now substitute these expanded forms back into the original expression:

becomes

Next, distribute the minus sign across the second parentheses.

Distribute the minus sign in front of $(5p^2 - 4pq - q^2)$:

Now group like terms:

• $p^2$ terms: $5p^2 - 5p^2$
• $pq$ terms: $13pq + 4pq + 20pq$
• $q^2$ terms: $-6q^2 + q^2 - 4q^2$

Combine each group:

• $5p^2 - 5p^2 = 0$
• $13pq + 4pq + 20pq = 37pq$
• $-6q^2 + q^2 - 4q^2 = -9q^2$.

Putting it all together, the original expression simplifies to

This matches answer choice A, $37pq - 9q^2$.