Question: Expand the product $ (5a - 3b)(2a + 7b) $.

["Title: Expanding the Product $ (5a - 3b)(2a + 7b) $: Step-by-Step Guide", "---", "Meta Description:\nLearn how to expand the binomial expression $ (5a - 3b)(2a + 7b) $ using the distributive property. Discover the full expansion and practical applications in algebra.", "---", "### Introduction", "Expanding algebraic expressions is a foundational skill in algebra, essential for solving equations, simplifying formulas, and tackling more complex math problems. One common operation is expanding a product of two binomials, such as $ (5a - 3b)(2a + 7b) $. In this article, we’ll walk through the process step-by-step, explain the underlying concept — often referred to informally as “expanding the product” — and show how this technique applies broadly across algebra.", "---", "### What Does “Expand the Product” Mean?", "When we say “expand the product $ (5a - 3b)(2a + 7b) $”, we mean multiplying every term in the first binomial by every term in the second binomial, then combining like terms. This is based on the distributive property, also known as the FOIL method (First, Outer, Inner, Last), which ensures every pair of terms is multiplied.", "---", "### Step-by-Step Expansion", "Let’s expand $ (5a - 3b)(2a + 7b) $ systematically:", "1. Apply the distributive property:\nMultiply each term in $ (5a - 3b) $ by each term in $ (2a + 7b) $:", "$$\n (5a)(2a) + (5a)(7b) + (-3b)(2a) + (-3b)(7b)\n $$", "2. Calculate each product individually:", "- $ 5a \cdot 2a = 10a^2 $\n - $ 5a \cdot 7b = 35ab $\n - $ -3b \cdot 2a = -6ab $\n - $ -3b \cdot 7b = -21b^2 $", "3. Write down all the terms:", "$$\n 10a^2 + 35ab - 6ab - 21b^2\n $$", "4. Combine like terms:", "The terms $ 35ab $ and $ -6ab $ are like terms (both contain $ ab $):", "$$\n 10a^2 + (35ab - 6ab) - 21b^2 = 10a^2 + 29ab - 21b^2\n $$", "---", "### Final Result", "$$\n(5a - 3b)(2a + 7b) = 10a^2 + 29ab - 21b^2\n$$", "---", "### Practical Applications", "Understanding how to expand such expressions opens the door to solving quadratic equations, simplifying rational expressions, and working with coordinate geometry (e.g., expanding area formulas). Mastering this technique strengthens algebraic reasoning and prepares students for higher-level math.", "---", "### Conclusion", "Expanding $ (5a - 3b)(2a + 7b) $ begins with carefully applying the distributive property across both binomials, then simplifying the result by combining like terms. Follow these clear steps, and you’ll confidently expand and simplify binomial products every time. Keep practicing — algebra gets easier with each expansion!", "---", "Ready to expand more? Practice with other binomials like $ (x - 4)(x + 3) $ or $ (2y + 5)(y - 1) $ and master this essential algebra skill today!", "---", "Keywords: expand product $ (5a - 3b)(2a + 7b) $, algebra tutoring, binomial expansion, distributive property, algebraic simplification, expand binomials, school math, high school algebra, mathematical formulas.", "---", "Additional Tips:\n- Always list all terms before combining.\n- Watch for sign errors — easy to miss with negative signs.\n- Practice expanding with different coefficients and variables.\n- Use area models or grid diagrams to visualize the expansion.", "---", "By mastering expansion techniques, you build a solid foundation for algebraic fluency. Start expanding—your math skills are stronger than ever!"]