What is the value of ( (3p+q)^2-(3p-q)^2-12pq )?

(0)

This expression uses the difference-of-squares identity. For any two quantities, \\(a+b\\)^2-\\(a-b\\)^2=4ab. Here, the two quantities are \\(a=3p\\) and \\(b=q\\). Therefore, the difference between the first two squared expressions is exactly \\(4(3p)(q)=12pq\\). The final term in the expression is also \\(12pq\\), so it cancels this difference completely. Thus, the value is zero, making option A correct. The result does not depend on particular values of \\(p\\) and \\(q\\); it follows algebraically for all values.

Expanding also verifies the result directly: \\( (3p+q)^2=9p^2+6pq+q^2\\) and \\( (3p-q)^2=9p^2-6pq+q^2\\). Subtracting the second from the first gives \\(12pq\\), because the \\(9p^2\\) and \\(q^2\\) terms cancel while the middle terms add. The remaining subtraction is \\(12pq-12pq=0\\). Option C is only the difference of the first two squares, not the value of the complete expression.

This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Algebraic identities.