1. **State the problem:** Find the least common multiple (LCM) of the polynomials $9(x + 2)(2x - 1)$ and $3(x + 2)$.\n\n2. **Recall the formula and rules:** The LCM of two polynomials is the polynomial of least degree that is divisible by both. To find it, factor each polynomial completely and take the product of the highest powers of all factors appearing in either polynomial.\n\n3. **Factor each polynomial:**\n- First polynomial: $9(x + 2)(2x - 1)$ is already factored.\n- Second polynomial: $3(x + 2)$ is also factored.\n\n4. **Identify common and unique factors:**\n- Common factor: $(x + 2)$\n- Unique factors: $9$ and $(2x - 1)$ in the first, $3$ in the second.\n\n5. **Compare coefficients 9 and 3:**\n- $9 = 3 \times 3$\n- The highest power of 3 appearing is $9$ (which is $3^2$).\n\n6. **Construct the LCM:**\n- Take the highest coefficient: $9$\n- Include the common factor $(x + 2)$ once\n- Include the unique factor $(2x - 1)$ from the first polynomial\n\nSo, $$\text{LCM} = 9(x + 2)(2x - 1)$$\n\n7. **Final answer:** The least common multiple of $9(x + 2)(2x - 1)$ and $3(x + 2)$ is $$9(x + 2)(2x - 1)$$.