Question: Expand the product $(2x - 3y)(x^2 + 4xy - 5y^2)$.
["# Expanding the Product $(2x - 3y)(x^2 + 4xy - 5y^2)$: A Step-by-Step Guide", "Multiplication of polynomials is a fundamental algebraic skill that plays a key role in expanding expressions for calculus, physics, and engineering applications. In this comprehensive guide, we focus on expanding the expression $(2x - 3y)(x^2 + 4xy - 5y^2)$ using the distributive property (also known as the FOIL method for binomials). Understanding how to expand this product thoroughly prepares you for solving more complex algebraic problems.", "---", "## Step 1: Understand the Distribution", "To expand $(2x - 3y)(x^2 + 4xy - 5y^2)$, we apply the distributive property:", "> Every term inside the first parentheses multiplies each term inside the second parentheses.", "This means:\n$$\n(2x - 3y)(x^2 + 4xy - 5y^2) = 2x(x^2 + 4xy - 5y^2) - 3y(x^2 + 4xy - 5y^2)\n$$", "---", "## Step 2: Multiply $2x$ Across the Second Polynomial", "Distribute $2x$ to each term inside $(x^2 + 4xy - 5y^2)$:", "$$\n2x \cdot x^2 = 2x^3\n$$\n$$\n2x \cdot 4xy = 8x^2y\n$$\n$$\n2x \cdot (-5y^2) = -10xy^2\n$$", "So, the contribution from the first part is:\n$$\n2x^3 + 8x^2y - 10xy^2\n$$", "---", "## Step 3: Multiply $-3y$ Across the Second Polynomial", "Now distribute $-3y$ to each term inside $(x^2 + 4xy - 5y^2)$:", "$$\n-3y \cdot x^2 = -3x^2y\n$$\n$$\n-3y \cdot 4xy = -12xy^2\n$$\n$$\n-3y \cdot (-5y^2) = 15y^3\n$$", "So, the contribution from the second part is:\n$$\n-3x^2y - 12xy^2 + 15y^3\n$$", "---", "## Step 4: Combine All Terms", "Now combine the results from both parts:", "$$\n2x^3 + 8x^2y - 10xy^2 - 3x^2y - 12xy^2 + 15y^3\n$$", "Group like terms:", "- $x^3$: $2x^3$\n- $x^2y$: $8x^2y - 3x^2y = 5x^2y$\n- $xy^2$: $-10xy^2 - 12xy^2 = -22xy^2$\n- $y^3$: $15y^3$", "---", "## Final Expanded Expression:", "$$\n\boxed{2x^3 + 5x^2y - 22xy^2 + 15y^3}\n$$", "---", "## Why This Expansion Matters", "Expanding expressions like $(2x - 3y)(x^2 + 4xy - 5y^2)$ helps build intuition for polynomial operations, factor expressions, and prepare for derivatives or integrals in higher mathematics. Mastering such expansions ensures a solid foundation in algebra.", "---", "### Summary", "- Use the distributive property to expand binomials over trinomials.\n- Carefully multiply each term and collect like terms.\n- The final result is $2x^3 + 5x^2y - 22xy^2 + 15y^3$.", "If you want to deepen your skills in polynomial multiplication and algebra, practice with varied examples—consistent effort leads to confidence and clarity!"]
