P = {x:(x - 2)(x + 3) (x - 4)(x + 5) = 0}. The elements in P are

Analyze the question: P is the set of all values of x that make the product equal to zero. We need to find every such value.

Concept: The Zero Product Property says that if a product of factors equals zero, then at least one factor must equal zero. So we set each factor equal to zero and solve.

Solution:

(x − 2)(x + 3)(x − 4)(x + 5) = 0

• x − 2 = 0 → x = 2
• x + 3 = 0 → x = −3
• x − 4 = 0 → x = 4
• x + 5 = 0 → x = −5

So P = {−5, −3, 2, 4}.

Check against the options:

• A (−5, −2, 3, 4): has −2 and 3, which are wrong signs
• B (−5, −3, 2, 4): matches exactly
• C (−5, −3, −2, 4): −2 is wrong
• D (−4, −3, −2, 5): wrong
• E (2, 3, 4, 5): all signs are wrong

Tip: For a factor like (x − a), the root is +a. For (x + a), the root is −a. The sign flips, so a quick way to avoid mistakes is to ask, "What value of x makes this bracket zero?"