Question: Expand the product $(2x^2 - 3x + 5)(x - 4)$.
["# Expanding the Product: $(2x^2 - 3x + 5)(x - 4)$", "Understanding how to expand polynomial expressions is a fundamental skill in algebra. One common task involves multiplying two polynomials, such as $(2x^2 - 3x + 5)(x - 4)$. Expanding this expression not only strengthens algebraic manipulation abilities but also helps in simplifying equations and solving higher-order problems. In this article, we’ll walk through the step-by-step expansion and provide clear explanations for each stage.", "## Understanding the Expression", "Before expanding, we start with the given expression:", "[\n(2x^2 - 3x + 5)(x - 4)\n]", "This is the product of a quadratic polynomial $(2x^2 - 3x + 5)$ and a linear polynomial $(x - 4)$. The distributive property (also known as the FOIL method for binomials) allows us to expand such expressions by multiplying every term in the first polynomial by every term in the second.", "## Step-by-Step Expansion", "### Step 1: Distribute each term in $(x - 4)$ across $(2x^2 - 3x + 5)$", "[\n(2x^2 - 3x + 5)(x - 4) = (2x^2 - 3x + 5) \cdot x + (2x^2 - 3x + 5) \cdot (-4)\n]", "Now distribute $x$ and $-4$ separately.", "---", "### Step 2: Multiply $x$ by each term in $(2x^2 - 3x + 5)$", "[\nx \cdot 2x^2 = 2x^3\n]\n[\nx \cdot (-3x) = -3x^2\n]\n[\nx \cdot 5 = 5x\n]", "So, the result of distributing $x$ is:\n[\n2x^3 - 3x^2 + 5x\n]", "---", "### Step 3: Multiply $-4$ by each term in $(2x^2 - 3x + 5)$", "[\n-4 \cdot 2x^2 = -8x^2\n]\n[\n-4 \cdot (-3x) = 12x\n]\n[\n-4 \cdot 5 = -20\n]", "So, the result of distributing $-4$ is:\n[\n-8x^2 + 12x - 20\n]", "---", "### Step 4: Combine the results", "Now, add the two polynomials we obtained:", "[\n(2x^3 - 3x^2 + 5x) + (-8x^2 + 12x - 20)\n]", "Group like terms:", "- $2x^3$ (only cubic term)\n- $-3x^2 - 8x^2 = -11x^2$\n- $5x + 12x = 17x$\n- $-20$ (constant term)", "---", "### Final Expanded Expression", "[\n(2x^2 - 3x + 5)(x - 4) = 2x^3 - 11x^2 + 17x - 20\n]", "## Summary and Key Takeaway", "Expanding $(2x^2 - 3x + 5)(x - 4)$ involves distributing each term across the second polynomial and combining like terms. The final expanded form is:", "[\n\boxed{2x^3 - 11x^2 + 17x - 20}\n]", "Mastering this technique builds a solid foundation for more advanced algebra, such as simplifying rational expressions, solving polynomial equations, and graphing functions. Practice this method with other polynomials to gain confidence and fluency.", "If you're preparing for exams or tackling algebra challenges, understanding product expansion step-by-step is key to success—start by applying the distributive property, combine terms carefully, and verify your work through substitution."]
