Question: Expand the product $ (2a - 3b)(4a^2 + 6ab + 9b^2) $.
["# Expanding the Expression: $ (2a - 3b)(4a^2 + 6ab + 9b^2) $", "Understanding the Product of a Binomial and a Sum of Cubes", "When multiplying the expression $ (2a - 3b) $ by $ (4a^2 + 6ab + 9b^2) $, students and algebra learners often recognize a key algebraic identity: the difference of cubes formula. While this expression resembles a substitution pattern for $ (x - y)(x^2 + xy + y^2) = x^3 - y^3 $, the coefficients here make it a subtle variation that’s crucial to expand carefully.", "## The Idea Behind the Identity", "Recall the standard identity:", "$$\n(x - y)(x^2 + xy + y^2) = x^3 - y^3\n$$", "If $ x = 2a $ and $ y = 3b $, then:", "- $ x - y = 2a - 3b $\n- $ x^2 + xy + y^2 = (2a)^2 + (2a)(3b) + (3b)^2 = 4a^2 + 6ab + 9b^2 $", "Thus, $ (2a - 3b)(4a^2 + 6ab + 9b^2) $ fits the first factor form of the difference of cubes identity.", "---", "## Expanding Step-by-Step Using the Distributive Property", "To confirm and understand this fully, let’s expand using the distributive property:", "$$\n\begin{align}\n(2a - 3b)(4a^2 + 6ab + 9b^2) &= 2a(4a^2 + 6ab + 9b^2) - 3b(4a^2 + 6ab + 9b^2) \\n&= 2a \cdot 4a^2 + 2a \cdot 6ab + 2a \cdot 9b^2 - 3b \cdot 4a^2 - 3b \cdot 6ab - 3b \cdot 9b^2 \\n&= 8a^3 + 12a^2b + 18ab^2 - 12a^2b - 18ab^2 - 27b^3 \\n\end{align}\n$$", "---", "## Combine Like Terms", "Now, combine the like terms:", "- $ 8a^3 $\n- $ 12a^2b - 12a^2b = 0 $\n- $ 18ab^2 - 18ab^2 = 0 $\n- $ -27b^3 $", "Thus, the final expanded form is:", "$$\n8a^3 - 27b^3\n$$", "---", "## Final Answer", "$$\n\boxed{(2a - 3b)(4a^2 + 6ab + 9b^2) = 8a^3 - 27b^3}\n$$", "This result confirms the difference of cubes factorization:", "$$\n(x - y)(x^2 + xy + y^2) = x^3 - y^3\n$$", "applied with $ x = 2a $, $ y = 3b $, yielding $ x^3 - y^3 = 8a^3 - 27b^3 $", "---", "## Why This Matters – Benefits of Understanding This Expansion", "Recognizing and expanding such patterns strengthens algebraic fluency and helps solve more complex polynomial identities. It’s essential for mastering factoring, simplifying radicals, and preparing for higher-level math like calculus and algebra in advanced courses.", "Remember: When you see a binomial times a trinomial resembling $ x^2 + xy + y^2 $, think $ x^3 - y^3 $. This powerful technique saves time and avoids mistake-prone full expansion.", "---", "Keywords: expand $ (2a - 3b)(4a^2 + 6ab + 9b^2) $, difference of cubes, algebraic identities, polynomial expansion, simplify expressions, math practice, algebra tutorial."]
