Factorize the expression: am + bn - an - bm

Analyze the question: We need to factorize a four-term expression, am + bn - an - bm. This is done by grouping.

Concept: Group terms that share a common factor, factor each group, then pull out the common bracket.

Solution:

Rearrange so terms with common factors sit together:

am - bm - an + bn

Group in pairs:

(am - bm) + (-an + bn)

Factor each group:

• m(a - b) from the first pair
• -n(a - b) from the second pair, since -an + bn = -n(a - b)

Now the common bracket is (a - b):

m(a - b) - n(a - b) = (a - b)(m - n)

Check by expanding:

(a - b)(m - n) = am - an - bm + bn

This matches the original expression am + bn - an - bm. ✓

Answer: Option B, (a - b)(m - n).

Tip: For four-term factorization, rearrange so that each pair shares a common factor and the brackets come out identical. If the brackets are not identical, check your signs. When you factor out a negative, the signs inside the bracket flip. Always expand your answer quickly to confirm.