Numerator: \( (3 + 2i)(4 + i) = 12 + 3i + 8i + 2i^2 = 12 + 11i - 2 = 10 + 11i \)

["Understanding Complex Number Multiplication: A Step-by-Step Guide with ( (3 + 2i)(4 + i) = 10 + 11i )", "Complex numbers may look intimidating at first, but their multiplication follows a clear, logical process rooted in basic algebra and the rules of imaginary numbers. One common example is finding the product ( (3 + 2i)(4 + i) ), which yields ( 10 + 11i ). In this article, we break down the computation step-by-step and explain the core principles behind complex number multiplication—helping you master this foundational topic for better understanding of complex arithmetic.", "---", "### What Are Complex Numbers?", "A complex number combines a real part and an imaginary part, represented as ( a + bi ), where ( i ) is the imaginary unit defined by ( i^2 = -1 ). Numbers like ( 3 + 2i ) or ( 4 + i ) are typical complex numbers used in mathematics, physics, and engineering.", "---", "### Why Multiply Complex Numbers?", "Multiplication of complex numbers extends the algebra of real numbers, enabling solutions to equations involving imaginary variables. It is crucial in fields like electrical engineering, signal processing, and quantum mechanics. Understanding the expansion ( (3 + 2i)(4 + i) ) provides insight into how real and imaginary parts interact.", "---", "### Step-by-Step Computation of ( (3 + 2i)(4 + i) )", "Start with the expression:\n[\n(3 + 2i)(4 + i)\n]", "Use the distributive property (also known as FOIL in algebra):\n- Multiply first terms: ( 3 \ imes 4 = 12 )\n- Outer terms: ( 3 \ imes i = 3i )\n- Inner terms: ( 2i \ imes 4 = 8i )\n- Last terms: ( 2i \ imes i = 2i^2 )", "Putting it together:\n[\n(3 + 2i)(4 + i) = 12 + 3i + 8i + 2i^2\n]", "---", "### Simplifying Using ( i^2 = -1 )", "Since ( i^2 = -1 ), replace ( 2i^2 ) with ( 2(-1) = -2 ):\n[\n12 + 3i + 8i - 2\n]", "Now combine like terms:\n- Real parts: ( 12 - 2 = 10 )\n- Imaginary parts: ( 3i + 8i = 11i )", "Thus, the final result is:\n[\n10 + 11i\n]", "---", "### Why the Expansion ( 12 + 3i + 8i + 2i^2 = 10 + 11i ) Matters", "- Real-world application: This type of calculation appears in complex impedance, AC circuit analysis, and wave functions.\n- Foundational skill: Mastering expansion and simplification prepares you for more advanced operations involving complex numbers, such as division or solving quadratic equations with imaginary roots.\n- Error prevention: Careful handling of ( i^2 ) avoids sign mistakes common among beginners.", "---", "### Final Thoughts", "Multiplying complex numbers like ( (3 + 2i)(4 + i) = 10 + 11i ) is a straightforward application of distributive rules and algebraic simplification. Remembering that ( i^2 = -1 ) is key to reducing the expression to a standard complex form. With practice, complex multiplication becomes intuitive, opening doors to deeper mathematical and scientific problem-solving.", "---", "Keywords: complex numbers, multiplication of complex numbers, ( (3 + 2i)(4 + i) = 10 + 11i ), imaginary unit ( i ), ( i^2 = -1 ), complex arithmetic, FOIL method, algebraic simplification.", "---", "Enhance your complex number skills today and never fear the next time you see ( (a + bi)(c + di) = ? )! With clear steps and practice, you’ll master any product."]