First, compute \( (4 + 2i)^2 \):

["# How to Compute ( (4 + 2i)^2 ): A Step-by-Step Guide with Full Explanation", "Original title: Compute ( (4 + 2i)^2 ): Step-by-Step Calculation & Complex Number Multiplication Guide", "Studying complex numbers can feel overwhelming at first, especially when squaring expressions like ( (4 + 2i)^2 ). But don’t worry—this comprehensive guide breaks down the computation step-by-step, helping you master complex arithmetic with confidence. Whether you're a student, educator, or math enthusiast, learning how to compute ( (4 + 2i)^2 ) opens the door to deeper complex number operations.", "---", "## What is a Complex Number?", "A complex number is written in the form:\n[\na + bi\n]\nwhere ( a ) is the real part, ( b ) is the imaginary part, and ( i = \sqrt{-1} ).", "In this case:\n[\n4 + 2i \quad \ ext{where} \quad a = 4, \quad b = 2\n]", "---", "## Step 1: Apply the Square Formula\nTo compute ( (4 + 2i)^2 ), we use the identity:\n[\n(z)^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2\n]\nThis expands into:\n[\n(4 + 2i)^2 = 4^2 + 2 \cdot 4 \cdot 2i + (2i)^2\n]", "---", "## Step 2: Compute Each Term\nBreak the expression into three parts:", "- First term:\n[\n4^2 = 16\n]", "- Second term (imaginary part):\n[\n2 \cdot 4 \cdot 2i = 16i\n]", "- Third term (using ( i^2 = -1 )):\n[\n(2i)^2 = 2^2 \cdot i^2 = 4 \cdot (-1) = -4\n]", "---", "## Step 3: Combine All Terms\nNow add the results:\n[\n(4 + 2i)^2 = 16 + 16i - 4\n]", "Combine the real parts:\n[\n16 - 4 = 12\n]", "So:\n[\n(4 + 2i)^2 = 12 + 16i\n]", "---", "## Why This Method Works\nUsing the binomial expansion ( (a + bi)^2 = a^2 + 2abi + (bi)^2 ) simplifies work and reduces errors. Remembering ( i^2 = -1 ) is key to simplifying the final term.", "---", "## Applications of ( (4 + 2i)^2 ) and Complex Squaring\nSquaring complex numbers like ( (4 + 2i)^2 = 12 + 16i ) appears in:\n- Electrical engineering (AC circuit analysis with impedance)\n- Signal processing (Fourier transforms)\n- Quantum mechanics (wave function calculations)\n- Fractional calculus and complex dynamics", "---", "## Summary\n- ( (4 + 2i)^2 ) computed step-by-step gives:\n[\n\boxed{12 + 16i}\n]\n- Use the formula ( (a + bi)^2 = a^2 - b^2 + 2abi ) for efficiency\n- Practice enhances fluency with complex arithmetic and its real-world applications", "---", "## Further Reading & Practice\n- Understand complex conjugates and modulus calculations\n- Try squaring other complex numbers like ( (3 - i)^2 ) and ( (1 + 3i)^2 )\n- Explore how complex number squaring behaves geometrically in the complex plane", "---", "Keywords: compute ( (4 + 2i)^2 ), complex number squaring, complex arithmetic, binomial expansion complex, imaginary number ( i ), mathematical skills, algebra, high school math, college-level complex numbers", "Meta Description:\nLearn how to compute ( (4 + 2i)^2 ) step-by-step with full explanation, real-world applications, and FAQs for students mastering complex numbers."]