In this aptitude (simplification and algebra) question, simplify the algebraic fraction: (4a^2 - 9b^2) / (2a + 3b), by recognising any standard factorisation pattern, and choose the correct simplified expression.

Introduction / Context:
This problem checks the ability to recognise and use the difference of squares identity to simplify an algebraic fraction. Such pattern recognition is extremely useful in algebra and helps in quickly simplifying expressions that may otherwise appear complex.

Given Data / Assumptions:
We must simplify (4a^2 - 9b^2) / (2a + 3b).
Variables a and b are assumed to be such that the denominator 2a + 3b is not zero.

Concept / Approach:
The numerator 4a^2 - 9b^2 can be viewed as a difference of squares: (2a)^2 - (3b)^2. The identity for difference of squares is u^2 - v^2 = (u - v)(u + v). Once the numerator is factorised using this pattern, we can cancel any common factor with the denominator, as long as it is not zero, to obtain a simplified expression.

Step-by-Step Solution:
Start with the numerator: 4a^2 - 9b^2. Recognise this as (2a)^2 - (3b)^2. Using the identity u^2 - v^2 = (u - v)(u + v), write 4a^2 - 9b^2 = (2a - 3b)(2a + 3b). So the fraction becomes ( (2a - 3b)(2a + 3b) ) / (2a + 3b). Provided 2a + 3b is not zero, we can cancel this common factor from numerator and denominator. The simplified result is 2a - 3b.

Verification / Alternative check:
We can test the simplification by substituting simple numeric values for a and b that keep 2a + 3b non zero. For example, let a = 2 and b = 1. The original expression is (4*4 - 9*1) / (2*2 + 3*1) = (16 - 9) / (4 + 3) = 7/7 = 1. The simplified expression 2a - 3b becomes 2*2 - 3*1 = 4 - 3 = 1. Since both give the same numeric value, the simplification is consistent.

Why Other Options Are Wrong:
Option 2a + 3b is one factor of the numerator, not the simplified quotient. Options 2a and 3b ignore part of the expression and do not arise from correct factorisation and cancellation. Option 4a - 9b is just the original numerator without division. Only 2a - 3b represents the correct simplified form of the given fraction.

Common Pitfalls:
Some learners try to cancel terms directly from 4a^2 - 9b^2 and 2a + 3b without factorising, which is not valid. Another common error is to misidentify the difference of squares and write (4a - 9b)(4a + 9b), which is incorrect. Recognising that 4a^2 is (2a)^2 and 9b^2 is (3b)^2 is essential.

Final Answer:
The simplified form of (4a^2 - 9b^2) / (2a + 3b) is 2a - 3b.