If the expression \( (3x - 4)(2x + 5) \) is expanded, what is the resulting quadratic expression?

["# Simplifying the Product: Expanding ( (3x - 4)(2x + 5) ) and Understanding the Resulting Quadratic Expression", "When faced with the algebraic expression ( (3x - 4)(2x + 5) ), many students wonder how to expand it and what the final quadratic form looks like. Expanding binomial products using the distributive property (also known as FOIL method) not only reveals the correct quadratic expression but also strengthens foundational algebra skills. In this article, we’ll walk through the step-by-step process of expanding ( (3x - 4)(2x + 5) ), explain the logic behind each step, and clarify the resulting quadratic expression.", "## What Happens When You Expand ( (3x - 4)(2x + 5) )?", "The expression ( (3x - 4)(2x + 5) ) is a multiplication of two binomials. To expand it, we apply the distributive property:\nEach term in the first binomial must be multiplied by each term in the second binomial.", "### Step 1: Apply the FOIL Method", "FOIL stands for First, Outer, Inner, Last — a helpful memory for tracking terms during multiplication:", "- First: Multiply the first terms in each binomial:\n ( 3x \cdot 2x = 6x^2 )", "- Outer: Multiply the outer terms:\n ( 3x \cdot 5 = 15x )", "- Inner: Multiply the inner terms:\n ( -4 \cdot 2x = -8x )", "- Last: Multiply the last terms:\n ( -4 \cdot 5 = -20 )", "### Step 2: Combine All Products", "Add all the partial results together:\n[\n6x^2 + 15x - 8x - 20\n]", "### Step 3: Simplify Like Terms", "Combine the linear terms ( 15x - 8x = 7x ). The constant ( -20 ) remains unchanged.", "### Final Result", "The expanded and simplified quadratic expression is:\n[\n\boxed{6x^2 + 7x - 20}\n]", "This is a standard quadratic in the form ( ax^2 + bx + c ), where:\n- ( a = 6 )\n- ( b = 7 )\n- ( c = -20 )", "## Why Knowledge of Expansion Matters", "Understanding how to expand expressions like ( (3x - 4)(2x + 5) ) is essential for solving equations, factoring quadratics, and working with polynomial graphs. Recognizing the structure of quadratic expressions also supports advanced topics in algebra, calculus, and beyond.", "Whether you're a high school student mastering algebra or someone refreshing foundational math skills, knowing that ( (3x - 4)(2x + 5) = 6x^2 + 7x - 20 ) simplifies future problem-solving and builds confidence in mathematical expression manipulation.", "### Practice Tip", "Try expanding other binomial products using the FOIL method to solidify your understanding. For example:\nTry ( (x + 3)(x - 2) ), or ( (2x + 1)(3x - 5) ), to explore how signs and coefficients affect the outcome.", "---", "In summary, expanding and simplifying ( (3x - 4)(2x + 5) ) yields the quadratic expression ( \boxed{6x^2 + 7x - 20} ), a clear example of applying multiplication rules to discover structured polynomial forms."]