Factorize completely: 6ax - 12by - 9ay + 8bx

Analyze the question: We have four terms with no common factor across all of them, so we use factorization by grouping. The trick is to group terms that share a common factor.

Concept: Rearrange the terms so each pair has a common factor, then factor out the common binomial.

Solution:

Expression: 6ax - 12by - 9ay + 8bx

Grouping 6ax with -9ay (both contain a), and 8bx with -12by (both contain b):

6ax - 9ay + 8bx - 12by

• First pair: 3a(2x - 3y)
• Second pair: 4b(2x - 3y)

Check the second pair: 4b × 2x = 8bx ✓ and 4b × (-3y) = -12by ✓

Now factor out the common bracket (2x - 3y):

(2x - 3y)(3a + 4b)

Verification by expanding option B: (3a + 4b)(2x - 3y) = 6ax - 9ay + 8bx - 12by ✓

This matches the original expression exactly.

Why the others fail: Options A and D have a 4x or 2a/3b pattern that would give terms like 8ax or 4bx coefficients that do not match. Option C gives 6ax + 9ay, which has the wrong sign on the ay term.

Answer: B

Exam tip: When a four-term expression has no overall common factor, try pairing terms by shared letters. If the brackets in both pairs come out identical, you have grouped correctly. If they don't match, try a different pairing or reorder the terms. Always expand quickly to check.