Simplify the following expression. (3x + 17 − 2y)(3x − 17 − 2y)

Correct — D. The stem is an image: simplify (3x + 17 − 2y)(3x − 17 − 2y).

Both brackets share 3x − 2y and differ only in the sign before 17. Put u = 3x − 2y and the product becomes (u + 17)(u − 17), the a² − b² pattern:
= u² − 17² = (3x − 2y)² − 289

Expand the square, middle term 2 × 3x × 2y:
(3x − 2y)² = 9x² − 12xy + 4y²

So the product is 9x² − 12xy + 4y² − 289, the expression printed as option (d).

Why the others are wrong

• (a)
  Option (a) shows 9x² + 12xy + 4y² − 289, which is (3x + 2y)² − 289. Both brackets carry −2y, so the middle term is negative.
• (b)
  Option (b) shows 9x² − 6xy + 4y² − 289. The middle term of a square is 2 × 3x × 2y = 12xy; dropping the factor 2 halves it.
• (c)
  Option (c) shows 9x² − 12xy + 4y² + 289. The identity gives u² − b², so 17² is subtracted. A plus here undoes the whole point of the pattern.

The difference-of-squares identity (a + b)(a − b) = a² − b² does not care what a and b stand for. Here a is the whole binomial 3x − 2y and b is 17.

Spotting it means reading the two brackets against each other before multiplying anything: the parts that match become a, the part that flips sign becomes b.

Term-by-term expansion of four terms by four terms gets there too, but it takes four times as long and every sign is a chance to slip.

Key facts

• (a + b)(a − b) = a² − b², including when a and b are whole expressions.
• (3x − 2y)² = 9x² − 12xy + 4y², since the middle term is 2 × 3x × 2y.
• With a = 3x − 2y and b = 17, the product is (3x − 2y)² − 289.
• 17² = 289.

Common traps

• Multiplying all four terms out under time pressure instead of grouping 3x − 2y
• Dropping the factor 2 from the middle term of the square
• Losing the sign of −2y and expanding (3x + 2y)² instead

No directly related past PYQ was found.