Factorize completely: $x^2 - (y + z)^2$

Analyze the question: The expression x² − (y + z)² is a difference of two squares, so we can factorize it using a standard identity.

Concept: Difference of two squares: a² − b² = (a + b)(a − b)

Solution:

Let a = x and b = (y + z).

x² − (y + z)² = [x + (y + z)] [x − (y + z)]

Remove the inner brackets carefully, remembering the minus sign changes both signs in the second factor:

= (x + y + z)(x − y − z)

Check by expanding (optional): (x + y + z)(x − y − z) = x² − x(y + z) + x(y + z) − (y + z)² = x² − (y + z)² ✓

Matching the options:

• A. (x − y − z)(x − y − z): this is a perfect square, not a difference of squares. ✗
• B. (x + y + z)(x − y − z): matches our result. ✓
• C. (x + y + z)(x + y − z): wrong second factor. ✗
• D. (x + y − z)(x − y + z): this expands to x² − (y − z)², which is not the same. ✗

Tip: When you see a square minus a bracket squared, treat the whole bracket as a single term. Write (a + b)(a − b) first, then open the brackets, and watch the signs in the second factor, where the minus sign applies to everything inside the bracket.