Shortcut Trick
Identify the squares: a2, 4b2 = (2b)2, and 9c2 = (3c)2.
Check signs of cross-products: −4ab and −6ca are negative, while 12bc is positive.
Since 12bc is positive, b and c must have the same sign (both − or both +).
Since −4ab and −6ca are negative, a must have the opposite sign of b and c.
Testing signs: (a − 2b − 3c)2 gives −4ab, +12bc, and −6ca.
∴ The correct answer is (a − 2b − 3c)2.
Alternate Method
Given: The expression is a2 + 4b2 + 9c2 − 4ab + 12bc − 6ca.
Formula Used: The standard Trinomial Square Identity: (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx.
Comparing the given expression with the identity:
⇒ x2 = a2 → x = a
⇒ y2 = 4b2 → y = ±2b
⇒ z2 = 9c2 → z = ±3c
⇒ 2xy = −4ab ⇒ 2(a)(y) = −4ab ⇒ y = −2b
⇒ 2zx = −6ca ⇒ 2(z)(a) = −6ca ⇒ z = −3c
⇒ 2yz = 2(−2b)(−3c) = 12bc (Matches the given term)
⇒ Therefore, the factorized form is (x + y + z)2 = (a − 2b − 3c)2.
∴ The correct answer is (a − 2b − 3c)2.
Additional Information
Trinomial Expansion Rule
The expansion of (a + b + c)2 always results in 3 squared terms (always positive) and 3 cross-products. The signs of the cross-products depend entirely on the relative signs of a, b, and c.
Perfect Square Recognition
If an algebraic expression has the form x2 + y2 + z2 + 2xy + 2yz + 2zx, it is a perfect square of the trinomial (x + y + z).
Sign Analysis
In (a − b − c)2, only the terms involving 'a' multiplied by others (ab and ac) will be negative, while the product of two negatives (bc) becomes positive.