Solution: Apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 = 4x^2 - 12xy + 9y^2 \). oxed{4x^2 - 12xy + 9y^2}

["### The Power of the Square Formula: Master Equation Simplifying with ( (a - b)^2 = a^2 - 2ab + b^2 )", "Understanding fundamental algebraic formulas is a cornerstone of solving equations with confidence and accuracy. One of the most versatile and frequently used formulas is the expansion of the square of a binomial:\n[\n(a - b)^2 = a^2 - 2ab + b^2\n]\nThis simple identity unlocks the ability to simplify expressions efficiently, especially in algebra, calculus, and advanced problem-solving. In this article, we explore how applying this formula—particularly in the expression ( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 = 4x^2 - 12xy + 9y^2 )—demonstrates both precision and elegance in mathematical manipulation.", "---", "#### What is the Binomial Square Formula?", "The formula ( (a - b)^2 = a^2 - 2ab + b^2 ) lets you expand a squared binomial into a sum of squared terms minus twice the product of its components. This is immensely useful when cross-multiplication or simplification is required without multiple tedious multiplications.", "---", "#### Step-by-Step: Applying the Formula to the Expression", "Consider the expression:\n[\n(2x)^2 - 2 \cdot (2x) \cdot (3y) + (3y)^2\n]", "Let’s identify ( a = 2x ) and ( b = 3y ). Plugging into the formula:\n[\n(2x - 3y)^2 = (2x)^2 - 2 \cdot (2x)(3y) + (3y)^2\n]", "Now simplify each term individually:\n- ( (2x)^2 = 4x^2 )\n- ( 2 \cdot 2x \cdot 3y = 12xy ), so ( -2 \cdot (2x)(3y) = -12xy )\n- ( (3y)^2 = 9y^2 )", "Putting it all together:\n[\n(2x - 3y)^2 = 4x^2 - 12xy + 9y^2\n]", "This transformation confirms the formula’s power—simplifying complex terms into a compact, expanded form with clarity.", "---", "#### Real-World Applications of the Expansion", "1. Simplifying Quadratic Formulas", "In quadratic equations, completing the square often relies on recognizing perfect square trinomials derived from ( (a - b)^2 ), enabling easier root-finding.", "2. Polynomial Factorization", "Many polynomials, especially those appearing in optimization problems and geometric modeling, factor neatly using binomial squares.", "3. Calculus and Derivatives", "When differentiating expressions like ( (2x - 3y)^2 ), applying the chain rule becomes simpler after expanding the square.", "4. Geometry & Coordinate Planning", "In coordinate geometry, equations modeling parabolas or circular arcs frequently use binomial expansions for straightforward analysis.", "---", "#### Why Every Student and Professional Should Master This", "Practicing identities like ( (a - b)^2 ) and their expansions sharpens algebraic intuition, reduces calculation errors, and accelerates problem-solving across disciplines—from engineering to data science.", "---", "#### Conclusion", "The formula ( (a - b)^2 = a^2 - 2ab + b^2 ) isn’t just a rule—it’s a gateway to elegant, efficient math. Applying it to expressions like ( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 ) reliably transforms complex terms into a clean, expanded result:\n[\n\boxed{4x^2 - 12xy + 9y^2}\n]\nMastering this technique empowers deeper mastery of algebra and supports success in advanced mathematical pursuits.", "---", "Keywords: ( (a - b)^2 ), binomial expansion, algebra tips, expand ( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 ), simplify expression, algebra fundamentals, calculus algebra, mathematical identities.\nBoxed Result:\n[\n\boxed{4x^2 - 12xy + 9y^2}\n]"]