Area = (w + 3)(2w - 2) = 24.

["### Solving the Equation: Area = (w + 3)(2w - 2) = 24", "Understanding algebraic equations is essential in math, and equations involving area expressions offer a rich opportunity to sharpen problem-solving skills. Today, we explore how to solve the equation:", "Area = (w + 3)(2w - 2) = 24", "This problem centers around finding real values of variable ( w ) that satisfy the condition of the area being 24. Whether for geometry, physics, or real-world modeling, mastering such equations is key to unlocking deeper algebraic understanding.", "---", "### Step 1: Expand the Left Side", "Start by expanding the product on the left:", "[\n(w + 3)(2w - 2)\n]", "Use the distributive property (also known as the FOIL method):", "[\n= w \cdot 2w + w \cdot (-2) + 3 \cdot 2w + 3 \cdot (-2)\n]", "[\n= 2w^2 - 2w + 6w - 6\n]", "Combine like terms:", "[\n2w^2 + 4w - 6\n]", "So the equation becomes:", "[\n2w^2 + 4w - 6 = 24\n]", "---", "### Step 2: Simplify to Standard Quadratic Form", "Subtract 24 from both sides to set the equation to zero:", "[\n2w^2 + 4w - 6 - 24 = 0\n]", "[\n2w^2 + 4w - 30 = 0\n]", "To simplify, divide every term by 2:", "[\nw^2 + 2w - 15 = 0\n]", "---", "### Step 3: Solve the Quadratic Equation", "Now solve ( w^2 + 2w - 15 = 0 ) using factoring, completing the square, or the quadratic formula.", "Factoring Approach:", "Find two numbers that multiply to (-15) and add to (2). These are (5) and (-3):", "[\n(w + 5)(w - 3) = 0\n]", "Set each factor equal to zero:", "[\nw + 5 = 0 \quad \Rightarrow \quad w = -5\n]", "[\nw - 3 = 0 \quad \Rightarrow \quad w = 3\n]", "Check both solutions to ensure they are valid in the original context (area remains non-negative):", "- For ( w = -5 ): Area expression terms: ( w + 3 = -2 < 0 ), and ( 2w - 2 = -12 < 0 ), so area would be positive (negative × negative), but area cannot be conceptually negative — consider practical interpretation or domain restrictions.\n- For ( w = 3 ): ( w + 3 = 6 > 0 ) and ( 2w - 2 = 4 > 0 ) → valid area = ( 6 \cdot 4 = 24 ), valid.", "Thus, ( w = 3 ) is the meaningful solution.", "---", "### Step 4: Final Answer", "The solution to the equation\n(w + 3)(2w - 2) = 24", "is:", "[\n\boxed{w = 3}\n]", "---", "### Why This Equation Matters", "Equations modeling area help solve real-life problems such as maximizing space, calculating dimensions, or designing structures. Understanding how to expand, simplify, and solve quadratic form equations like this strengthens your foundation in algebra and applied mathematics.", "---", "### Summary", "- Expanded: ((w + 3)(2w - 2) = 24 \rightarrow 2w^2 + 4w - 6 = 24)\n- Simplified: (2w^2 + 4w - 30 = 0), then (w^2 + 2w - 15 = 0)\n- Solved via factoring: ( (w + 5)(w - 3) = 0 )\n- Valid solution: ( w = 3 )\n- Verified: ( Area = (3 + 3)(2 \cdot 3 - 2) = 6 \cdot 4 = 24 )", "Mastering these steps empowers you to tackle complex algebraic expressions with confidence.", "---", "For more advanced learning on solving polynomial equations and real-world applications, explore algebraic modeling in geometry and quadratic equations in practical contexts. Keep practicing — algebra is the gateway to deeper mathematical insight!"]