\[(x + 4)(x - 5) = x(x) + x(-5) + 4(x) + 4(-5).\]

["Unlocking the Power of Expanded Expressions: A Deep Dive into [(x + 4)(x - 5) = x^2 - 5x + 4x - 20]", "When tackling algebraic expressions, one of the most fundamental skills is expanding products—especially binomial expressions. A classic example is the expansion of [(x + 4)(x - 5)]. At first glance, multiplying these two binomials may seem intimidating, but applying the distributive property not only simplifies the problem but also reveals valuable algebraic structure.", "This article explores the expansion of [(x + 4)(x - 5)] and demonstrates how it connects to core algebraic principles like the distributive law, combining like terms, and the formation of standard quadratic forms.", "---", "Expanding the Expression Step by Step", "The expression\n[(x + 4)(x - 5)]\nis a product of two binomials. To expand it, we apply the distributive property (also known as FOIL—First, Outer, Inner, Last):", "1. First: (x \cdot x = x^2)\n2. Outer: (x \cdot (-5) = -5x)\n3. Inner: (4 \cdot x = 4x)\n4. Last: (4 \cdot (-5) = -20)", "Now, combine all these results:\n[x^2 - 5x + 4x - 20]", "Next, simplify by combining like terms:\n[-5x + 4x = -x]\nSo the fully expanded and simplified form is:\n[\boxed{x^2 - x - 20}]", "---", "Breaking Down the Expanded Form: [x^2 - x - 20]", "This expression reveals important algebraic structure. Let’s examine each part:", "- Quadratic term: (x^2) — the result of multiplying the linear terms (x) and (x)\n- Linear term: (-x) — the sum of the cross products: (x \cdot (-5) + 4 \cdot x = -5x + 4x)\n- Constant term: (-20) — the product of the constant terms: (4 \cdot (-5))", "This matches our earlier expansion:\n[(x + 4)(x - 5) = x^2 - x - 20]", "---", "Using the Distributive Property: Why It Works", "At the heart of this expansion is the distributive property of multiplication over addition:\n[a(b + c) = ab + ac]", "In [(x + 4)(x - 5)], we distribute (x) and then (4):", "[\n(x + 4)(x - 5) = x(x - 5) + 4(x - 5)\n]\n[= x \cdot x - 5x + 4x - 20 = x^2 - x - 20\n]", "This method not only confirms the correctness of the expansion but also reinforces a foundational algebraic technique—distributing terms precisely and systematically.", "---", "Why Expand [(x + 4)(x - 5]]", "You might wonder: Why go through the effort of expanding instead of just keeping it factored? The expanded form is essential for:", "- Solving equations: When solving [(x + 4)(x - 5) = 0], expanding leads to the standard quadratic form (x^2 - x - 20 = 0), which can be solved using factoring, completing the square, or the quadratic formula.\n- Graphing: The vertex form of a quadratic depends on the expanded and simplified version.\n- Understanding structure: Seeing how each term contributes deepens your grasp of algebraic operations and polynomial behavior.", "---", "Practice Makes Perfect: Try It Yourself", "Want to test this concept? Try expanding these similar expressions:\n- [(x + 2)(x - 3)]\n- [(2x + 1)(x - 4)]", "Each time, use the distributive property method to expand, then combine like terms. This practice strengthens fluency in algebraic manipulation.", "---", "Conclusion", "The expansion of [(x + 4)(x - 5) = x^2 - x - 20] beautifully illustrates the power of the distributive property in simplifying polynomial expressions. From distributing terms to combining like terms, each step builds toward a clearer, more useful form. Whether you’re solving equations, graphing parabolas, or mastering algebra fundamentals, mastering expansion is an indispensable skill.", "Next time you see [(x + a)(x + b)], remember: multiplying these binomials opens the door to a deeper understanding of quadratic expressions—and the confidence to tackle more complex algebra ahead.", "---", "Key Takeaways:\n- Expand [(x + 4)(x - 5)] using distribution:\n [(x + 4)(x - 5) = x^2 - x - 20]\n- Recognize the significance of combining like terms and identifying quadratic forms.\n- Practice with variations to strengthen algebra skills.", "Keywords: expand ((x + 4)(x - 5)), algebra, distributive property, polynomial expansion, quadratic expression, solving equations, algebraic manipulation."]