In this aptitude (simplification and algebraic identities) question, you are given that x - y = -9 and xy = -20. Using the identity relating x^2 + y^2, (x - y)^2, and xy, find the value of x^2 + y^2.

Introduction / Context:
This question tests the use of standard algebraic identities to compute expressions involving squares without solving for x and y individually. Such techniques are extremely useful in aptitude tests where time is limited and direct solution may be unnecessary.

Given Data / Assumptions:
We are given x - y = -9.
We are also given xy = -20.
We must find the value of x^2 + y^2.

Concept / Approach:
A useful identity is (x - y)^2 = x^2 + y^2 - 2xy. From this relationship we can solve for x^2 + y^2 in terms of (x - y)^2 and xy. Specifically, x^2 + y^2 = (x - y)^2 + 2xy. This allows us to plug in the given values directly without finding x and y separately.

Step-by-Step Solution:
Start with the identity (x - y)^2 = x^2 + y^2 - 2xy. Rearrange to express x^2 + y^2: x^2 + y^2 = (x - y)^2 + 2xy. We know x - y = -9, so (x - y)^2 = (-9)^2 = 81. We also know xy = -20, so 2xy = 2 * (-20) = -40. Now substitute into the formula: x^2 + y^2 = 81 + (-40). Therefore x^2 + y^2 = 81 - 40 = 41.

Verification / Alternative check:
As a rough check, we could attempt to find actual values of x and y by solving the quadratic formed from t^2 - (x + y)t + xy = 0, but this is not necessary. The identity based method is much faster and yields a clean integer result. Because the identity is standard and our substitutions are straightforward, the value 41 is reliable.

Why Other Options Are Wrong:
Option 61 would correspond to using x^2 + y^2 = (x - y)^2 - 2xy, which is the wrong sign in the formula. Option 51 and 21 are random alternatives that do not arise from correct application of the identity. Option 81 appears if we forget the 2xy term entirely and take only (x - y)^2. Since we correctly used x^2 + y^2 = (x - y)^2 + 2xy, only the result 41 matches.

Common Pitfalls:
A common mistake is to confuse the identities for (x + y)^2 and (x - y)^2 or to misplace the sign of the 2xy term. Another pitfall is to try solving for x and y directly, which wastes time and may introduce extra algebraic errors. Remembering the correct identity and using the given data efficiently is the best strategy.

Final Answer:
Using the identity and given values, we obtain x^2 + y^2 = 41.