= 3 \cdot 1 + 3 \cdot (-2i) + 4i \cdot 1 + 4i \cdot (-2i)

["# Simplifying and Solving the Complex Expression: 3·1 + 3·(−2i) + 4i·1 + 4i·(−2i)", "Hello math enthusiasts! Today, we’re diving into a key algebra concept—simplifying complex expressions involving imaginary numbers. We’ll break down the expression 3·1 + 3·(−2i) + 4i·1 + 4i·(−2i) step by step and explain how to solve it clearly and correctly. Whether you're learning complex numbers for the first time or sharpening your skills, this breakdown will help you master simple complex arithmetic.", "---", "## What You’ll Learn\n- How to simplify complex numbers step-by-step\n- Rules for combining real and imaginary parts\n- The importance of multiplying imaginary units correctly", "---", "## The Expression:\n3·1 + 3·(−2i) + 4i·1 + 4i·(−2i)", "---", "## Step 1: Identify Real and Imaginary Terms", "In complex arithmetic, the imaginary unit i satisfies i² = −1. We separate the expression into real and imaginary components by sorting terms accordingly.", "Original expression:\n3·1 + 3·(−2i) + 4i·1 + 4i·(−2i)", "Let’s expand each term:\n- ( 3 \cdot 1 = 3 ) (real)\n- ( 3 \cdot (−2i) = −6i ) (imaginary)\n- ( 4i \cdot 1 = 4i ) (imaginary)\n- ( 4i \cdot (−2i) = −8i² ) (imaginary × imaginary)", "Now substitute back:\n3 − 6i + 4i − 8i²", "---", "## Step 2: Simplify Using ( i^2 = −1 )", "Replace ( i^2 ) with −1 in the term −8i²:\n−8i² = −8(−1) = 8 (real part)", "Now substitute and rewrite the full expression:\n3 + 8 − 6i + 4i", "---", "## Step 3: Combine Like Terms", "Group real parts and imaginary parts:", "- Real part: ( 3 + 8 = 11 )\n- Imaginary part: ( −6i + 4i = −2i )", "Putting it together:\n11 − 2i", "---", "## Final Answer:\n3·1 + 3·(−2i) + 4i·1 + 4i·(−2i) = 11 − 2i", "---", "## Why This Matters", "Understanding how to simplify complex expressions is foundational in fields like electrical engineering, quantum physics, computer science, and complex signal processing. Mistakes with imaginary units often lead to calculation errors—so mastering the rules is essential.", "Whether you’re solving equations with complex numbers or checking homework, always simplify fully, check your arithmetic, and apply ( i^2 = −1 ) correctly.", "---", "## Recap\n- Expanded and grouped real and imaginary terms\n- Used ( i^2 = −1 ) to simplify\n- Final simplified form: 11 − 2i", "Keep practicing—complex numbers become easier every time you apply the rules consistently!", "---", "Keywords: complex numbers, simplify complex expression, imaginary unit i, i squared equals -1, algebra, mathematics tutorial, solve complex equations, imaginary arithmetic, complex number calculation, imaginary and real parts, math help for students", "Meta Title: How to Simplify the Complex Expression 3·1 + 3·(−2i) + 4i·1 + 4i·(−2i) Step by Step", "Meta Description: Learn to simplify and solve complex expressions involving i. Step-by-step guide with real and imaginary parts, plus example calculation of 3·1 + 3·(−2i) + 4i·1 + 4i·(−2i). Perfect for beginners and learners."]