Simplify; $\frac{2 - 18m^2}{1 + 3m}$

Analyze the question: We need to simplify a fraction whose numerator looks like a difference of two squares after we take out a common factor.

Concept: Difference of two squares: a² − b² = (a − b)(a + b)

Solution:

1. Factor out 2 from the numerator: 2 − 18m² = 2(1 − 9m²)

2. Rewrite 1 − 9m² as a difference of two squares: 1 − 9m² = (1)² − (3m)² = (1 − 3m)(1 + 3m)

3. So the numerator is 2(1 − 3m)(1 + 3m).

4. Divide by the denominator:

[2(1 − 3m)(1 + 3m)] / (1 + 3m) = 2(1 − 3m)

The (1 + 3m) cancels (for m ≠ −1/3).

Answer: 2(1 − 3m), which is option C.

Tip: When you see a subtraction with squares in the numerator and a bracket like (1 + 3m) in the denominator, check for a difference of two squares. Always take out the common factor first (here, 2), because it reveals the hidden squares. Then check your answer by expanding: 2(1 − 3m)(1 + 3m) = 2(1 − 9m²) = 2 − 18m². ✓