If xy(x + y) = m for real numbers x and y, then what is the simplified value of x³ + y³ + 3m in terms of x and y?

Introduction / Context:
This algebra question focuses on using standard identities for the sum of cubes. It checks whether you recognize how to rewrite x³ + y³ in terms of x + y and xy, and how to incorporate the given expression xy(x + y) = m into the identity. Mastering such identities is very helpful in simplification problems in aptitude and competitive exams.

Given Data / Assumptions:

• xy(x + y) = m.
• x and y are real numbers.
• We need x³ + y³ + 3m expressed in simplest form.
• Standard algebraic identities can be used.

Concept / Approach:
We recall the identity for the sum of cubes of two numbers: x³ + y³ = (x + y)³ − 3xy(x + y). If we then consider x³ + y³ + 3xy(x + y), the −3xy(x + y) and +3xy(x + y) cancel, leaving just (x + y)³. Since xy(x + y) is given as m, the term 3m is exactly 3xy(x + y). This observation allows a very quick simplification of the whole expression.

Step-by-Step Solution:
Start with the known identity: x³ + y³ = (x + y)³ − 3xy(x + y). We are asked to evaluate x³ + y³ + 3m. Given that m = xy(x + y), we have 3m = 3xy(x + y). Substitute the identity into the expression: x³ + y³ + 3m = [(x + y)³ − 3xy(x + y)] + 3xy(x + y). The terms −3xy(x + y) and +3xy(x + y) cancel out. So we get x³ + y³ + 3m = (x + y)³.

Verification / Alternative check:
Take a simple numerical example to validate. Let x = 1 and y = 2. Then x + y = 3 and xy = 2. Here m = xy(x + y) = 2·3 = 6. Now x³ + y³ + 3m = 1³ + 2³ + 3·6 = 1 + 8 + 18 = 27. On the other hand, (x + y)³ = 3³ = 27. Both sides are equal, confirming the correctness of the simplification for at least one concrete pair of values.

Why Other Options Are Wrong:
m³ would imply cubing xy(x + y), which is not supported by the identity. m(x + y) is dimensionally inconsistent with x³ + y³. The option (x³ + y³)³ is far larger power-wise and does not reduce to our expression. The plain x³ + y³ ignores the +3m term completely. Only (x + y)³ matches the identity-based simplification.

Common Pitfalls:
Students sometimes confuse the identity for x³ + y³ with that for x³ − y³, or they misremember the factor 3xy(x + y). Another common mistake is to attempt expansion from scratch instead of using the identity, which is longer and prone to error. Recognizing standard patterns like x³ + y³ + 3xy(x + y) saves time and reduces algebraic mistakes.

Final Answer:
Therefore, the simplified value of x³ + y³ + 3m is (x + y)³.