Simplify the algebraic expression by factorization: (4a^2 + 12ab + 9b^2) / (2a + 3b). Find the simplified result in lowest algebraic terms.

Introduction / Context:
This algebra problem asks you to simplify a rational expression by factorization. The numerator is a quadratic expression in a and b, and the denominator is a binomial. Recognizing common patterns such as perfect square trinomials helps you factor quickly and cancel common factors, which is a key skill in algebra and aptitude tests involving algebraic fractions.

Given Data / Assumptions:

• Expression: (4a^2 + 12ab + 9b^2) / (2a + 3b).
• a and b are real numbers such that the denominator (2a + 3b) is not zero.
• We must express the result in simplified form after factorization.

Concept / Approach:
Notice that the numerator 4a^2 + 12ab + 9b^2 resembles the expansion of a squared binomial. In particular, (2a + 3b)^2 = 4a^2 + 12ab + 9b^2. Once we factor the numerator into (2a + 3b)^2, we can simplify the fraction by canceling a common factor of (2a + 3b) in the numerator and denominator, assuming it is not zero. This process is a standard application of factorization and cancellation in rational expressions.

Step-by-Step Solution:
1) Consider the numerator: 4a^2 + 12ab + 9b^2. 2) Recognize the pattern of a perfect square: (2a)^2 = 4a^2, 2 * (2a) * (3b) = 12ab, and (3b)^2 = 9b^2. 3) Therefore 4a^2 + 12ab + 9b^2 = (2a + 3b)^2. 4) Substitute into the original expression: (4a^2 + 12ab + 9b^2) / (2a + 3b) = (2a + 3b)^2 / (2a + 3b). 5) Cancel the common factor (2a + 3b) in numerator and denominator, as long as (2a + 3b) ≠ 0. 6) The simplified expression is 2a + 3b.

Verification / Alternative check:
To verify, choose specific values for a and b, for example a = 1 and b = 1. Then the numerator becomes 4(1)^2 + 12(1)(1) + 9(1)^2 = 4 + 12 + 9 = 25. The denominator is 2(1) + 3(1) = 5. The fraction is 25 / 5 = 5. Evaluating 2a + 3b for a = 1 and b = 1 gives 2(1) + 3(1) = 5. The two results agree, confirming that the simplification is correct.

Why Other Options Are Wrong:
Option a (2a − 3b) would result if the middle term were negative, as in 4a^2 − 12ab + 9b^2, which is not the case here. Options c (2a) and d (3b) ignore one of the variables and do not align with the factorization. Option e (4a + 9b) comes from incorrectly combining coefficients rather than properly factoring. Only option b, 2a + 3b, matches the factorization of the numerator divided by the denominator.

Common Pitfalls:
A frequent mistake is misidentifying the perfect square structure or trying to factor the numerator into incorrect binomials. Some students also cancel terms instead of factors, which is algebraically invalid. Always factor the numerator completely first and cancel only common factors that multiply the entire numerator and denominator, not individual terms inside a sum.

Final Answer:
The simplified value of the expression is 2a + 3b.