Each statement will give us two solutions for x. We can then plug these two values of x in to the expression in the question, and if we get the same answer both times, the statement is sufficient, and if we get two different answers, the statement is not sufficient.

x^2 + 7x - 18 = 0

(x + 9)(x - 2) = 0

x = -9 or x = 2

If you plug these two values of x in to the question, you get two different answers, so Statement 1 is not sufficient.

Looking at Statement 2:

3x^2 + 46x + 171 = 0

(3x + something)(x + something else) = 0

Here the two "something" numbers must multiply to 171. Ordinarily we'd need to work out the divisors of 171, and possibly try out a few divisors until we found the pair that "works" (i.e. that gives "+46x" as the middle term if you multiply the two factors back together), which takes a long time and is the reason you never need to do this kind of factoring on the real GMAT. But we know the two Statements in a GMAT DS question must be consistent. There must be at least one value of x that works in both statements. Since x+9 and x-2 were the only factors in Statement 1, and since we want a factor that includes a number which is a divisor of 171, it must be that x+9 is a factor of the quadratic in Statement 2. So our factorization is

3x^2 + 46x + 171 = (3x + 19) (x + 9)

and just to be sure, we can multiply the right side back out to confirm this is correct. If we do that, we do get "46x" in the middle, so this is the correct factorization. So the two solutions for x are x = -9 and x = -19/3, and if you plug those back in to the question, you get an answer of 2 either way, so Statement 2 is sufficient alone.

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