Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is x + x^2 + x^3 > 0

(1) x+x^2>0

(2) x^2+x^3>0

In general, x^2+x+1>0 and x^2+1>0 are always valid. Also, when it comes to inequality DS questions, it is important that if range of que includes range of con, the con is sufficient.

When you modify the original condition and the question, 1+x+x^2>0 is always valid from x(1+x+x^2)>0?, which makes x>0? possible. Thus, there is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.

For 1), x(1+x)>0 -> x<-1, 0<x, the range of que doesn’t include the range of con, which is not sufficient.

For 2), when dividing x^2(1+x)>0 with x^2(since x^2 is positive, the sign of inequality doesn’t change when dividing with it.). Then, 1+x>0 becomes x>-1. The range of que doesn’t include the range of con, which is not sufficient.

When 1) & 2), from x>0, the range of que includes the range of con, which is sufficient.

Therefore, the answer is C.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

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