<p>If \(\dfrac{(x+2)(x-3)}{(x-3)(x-5)(x+5)^2} \leq 0\), the number of possible integral values of \(x\) is:</p>
Step-by-Step Solution
Key Concept: Cancel (x-3) but note x \neq 3. The (x+5)^2 factor is always non-negative. Use sign chart on remaining factors.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Cancel $(x-3)$ (valid since $x \neq 3$); $(x+5)^2 \geq 0$ always. Expression reduces to $\dfrac{x+2}{(x-5)(x+5)^2}\leq 0$. Critical points: $x=-5, -2, 5$. Sign chart gives solution intervals; count integers carefully. Total = 6 integral values. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: A