Question 3

(2x - 9)/((x + 2)(x + 3))

The expression x/(x ^ 2 + 6x + 9) - (x - 3)/(x ^ 2 + 5x + 6) simplifies to (2x - 9)/((x + 2)(x + 3)), demonstrating that this is the equivalent form.

A) (2x - 9)/((x + 2)(x + 3))

Option A is correct as it accurately represents the simplified version of the original expression. The denominators in both fractions can be factored, and upon performing the subtraction, the resulting numerator is confirmed to be 2x - 9.

B) (2x + 9)/((x + 2)(x + 3))

Option B is incorrect because the numerator does not correspond to the result of the subtraction. Instead of 2x - 9, this option incorrectly uses 2x + 9, which does not align with the simplification process of the expression.

C) (2x - 9)/((x + 2) * (x + 3) ^ 2)

Option C is incorrect as it alters the denominator by introducing an additional power to (x + 3) without justification from the original expression. The correct factorization does not require squaring (x + 3) in the denominator.

D) (2x + 9)/((x + 2) * (x + 3) ^ 2)

Option D is also incorrect for the same reason as Option B, where the numerator is incorrectly stated as 2x + 9, and the denominator has an unnecessary square of (x + 3). This does not reflect the correct simplification of the original fractions.

E) (2x + 9)/((x + 2) ^ 2 * (x + 3) ^ 2)

Option E is incorrect due to both a misrepresentation of the numerator as 2x + 9 and an unnecessary squaring of both factors in the denominator. The correct expression does not necessitate such alterations.

Conclusion

Option A is definitively correct as it accurately represents the simplified form of the original expression. All other options fail either by misrepresenting the numerator or incorrectly altering the structure of the denominator, demonstrating a lack of alignment with the mathematical simplification required.