What is the product of $2 p+q$ and $-3 q-6 p+1$?

Distribute 2 p and q across the terms in ( − 3 q − 6 p + 1 ) .
Multiply each term: 2 p ( − 3 q − 6 p + 1 ) + q ( − 3 q − 6 p + 1 ) = − 6 pq − 12 p 2 + 2 p − 3 q 2 − 6 pq + q .
Combine like terms: − 12 p 2 − 6 pq − 6 pq + 2 p − 3 q 2 + q = − 12 p 2 − 12 pq + 2 p − 3 q 2 + q .
The final answer is − 12 p 2 − 12 pq + 2 p − 3 q 2 + q ​ .

Explanation

Understanding the Problem We are asked to find the product of the expressions ( 2 p + q ) and ( − 3 q − 6 p + 1 ) . This involves multiplying each term in the first expression by each term in the second expression and then simplifying by combining like terms.

Applying the Distributive Property To find the product, we use the distributive property:


( 2 p + q ) ( − 3 q − 6 p + 1 ) = 2 p ( − 3 q − 6 p + 1 ) + q ( − 3 q − 6 p + 1 )

Distributing the Terms Now, we distribute 2 p and q across the terms in the parentheses:

2 p ( − 3 q − 6 p + 1 ) = − 6 pq − 12 p 2 + 2 p
q ( − 3 q − 6 p + 1 ) = − 3 q 2 − 6 pq + q

Combining Like Terms Next, we combine the two resulting expressions:

− 6 pq − 12 p 2 + 2 p − 3 q 2 − 6 pq + q = − 12 p 2 − 6 pq − 6 pq + 2 p − 3 q 2 + q
Combining like terms, we get:
− 12 p 2 − 12 pq + 2 p − 3 q 2 + q

Final Answer Comparing our simplified expression − 12 p 2 − 12 pq + 2 p − 3 q 2 + q with the given options, we find that it matches the second option:

− 12 p 2 − 12 pq + 2 p − 3 q 2 + q
Examples
Understanding how to multiply algebraic expressions is essential in various fields, such as physics and engineering. For instance, when calculating the area of a rectangle with sides defined by algebraic expressions, you need to multiply those expressions. Similarly, in physics, you might use this skill to determine the kinetic energy of an object, where both mass and velocity are expressed algebraically. This algebraic manipulation allows for precise calculations and predictions in real-world scenarios.

Answered by GinnyAnswer | 2025-07-03

The product of the expressions 2 p + q and − 3 q − 6 p + 1 is − 12 p 2 − 12 pq + 2 p − 3 q 2 + q . The answer is derived using the distributive property to multiply the terms and then combining like terms. The final answer is − 12 p 2 − 12 pq + 2 p − 3 q 2 + q ​ .
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Answered by Anonymous | 2025-07-04