Factorize 6pq-3rs-3ps+6qr

Analyze the question: We need to factorize 6pq - 3rs - 3ps + 6qr completely. This is factorization by grouping.

Concept: First take out the common factor (3), then group terms that share a common factor.

Solution:

Step 1: Take out the common factor 3:

6pq - 3rs - 3ps + 6qr = 3(2pq - rs - ps + 2qr)

Step 2: Rearrange so that terms with common factors sit together:

= 3(2pq + 2qr - ps - rs)

Step 3: Group in pairs:

= 3[2q(p + r) - s(p + r)]

Step 4: Take out the common bracket (p + r):

= 3(p + r)(2q - s)

Check by expanding: 3(p + r)(2q - s) = 3(2pq - ps + 2qr - rs) = 6pq - 3ps + 6qr - 3rs ✓

This matches the original expression.

Compare with the options:

• A. 3(r - p)(2q + s) expands to 3(2qr + rs - 2pq - ps), which is not equal to the original.
• B. 3(p + r)(2q - 2q - s) simplifies to 3(p + r)(-s), which is not equal to the original (it looks like a misprint).
• C. 3(2q - s)(p + r) is the same as our answer ✓
• D. 3(r - p)(s - 2q) expands to 3(rs - 2qr - ps + 2pq), which has the wrong signs on some terms.

Tip: In grouping questions, first pull out the number common to all terms, then rearrange so that the pairs share a variable. If the brackets that appear match, you are on the right track. Always verify by expanding quickly, since it takes only seconds and catches sign errors.