Data Sufficiency

p>q, Is q negative?

(1) pq + q^2 is positive
(2) (p^2)(q) + (p)(q^2) is positive

The OA is E.

How can I determine that the correct answer is the option E? May someone help me?

VJesus12 wrote:p>q, Is q negative?

(1) pq + q^2 is positive
(2) (p^2)(q) + (p)(q^2) is positive

Statement 1:
Case 1: q=1
Substituting q=1 into pq + q² > 0, we get:
p(1) + 1² > 0
p > -1.
Since this case requires that p>-1, and the prompt requires that p>q, it's possible that p=2.
Thus, a valid combination for Statement 1 is p=2 and q=1, in which case the answer to the question stem is NO.

Case 2: q=-1
Substituting q=-1 into pq + q² > 0, we get:
p(-1) + (-1)² > 0
-p > -1.
p < 1.
Since this case requires that p<1, and the prompt requires that p>q, it's possible that p=1/2.
Thus, a valid combination for Statement 1 is p=1/2 and q=-1, in which case the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statement 2:
Test whether Case 1 (p=2 and q=1) satisfies p²q + pq² > 0:
2²(1) + 2(1²) > 0
6 > 0.
This works.
Thus, a valid combination for Statement 2 is p=2 and q=1, in which case the answer to the question stem is NO..

Test whether Case 2 (p=1/2 and q=-1) satisfies p²q + pq² > 0:
(1/2)²(-1) + (1/2)(-1)² > 0
-1/4 + 1/2 > 0
1/4 > 0.
This works.
Thus, a valid combination for Statement 2 is p=1/2 and q=-1, in which case the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statements combined:
Cases 1 and 2 satisfy both statements.
Since the answer is NO in Case 1 but YES in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.