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Home » Whats Hidden in the Windows Icon Path? This Hidden Shortcut Every User Misses! - Imagemakers

Whats Hidden in the Windows Icon Path? This Hidden Shortcut Every User Misses! - Imagemakers

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 g(x^2 + 1) = 2(x^2 + 1)^2 - (x^2 + 1) - 2 = 2x^4 + 4x^2 + 2 - x^2 - 1 - 2 = 2x^4 + 3x^2 - 1.

📌 The final answer is $ \boxed{2x^4 + 3x^2 - 1} $.

📌 Question: An angel investor analyzes a medical startup's growth, modeled by the polynomial $ g(x) $, where $ g(x^2 + 2) = 3x^6 - 4x^4 + 5x^2 - 2 $. Find $ g(x^2 - 2) $.

📌 Solution: Let $ y = x^2 + 2 $. Then $ x^2 = y - 2 $. Express $ g(y) $ in terms of $ y $:

📌 g(y) = 3x^6 - 4x^4 + 5x^2 - 2 = 3(x^2)^3 - 4(x^2)^2 + 5x^2 - 2 = 3(y - 2)^3 - 4(y - 2)^2 + 5(y - 2) - 2.

📌 3(y^3 - 6y^2 + 12y - 8) - 4(y^2 - 4y + 4) + 5y - 10 - 2 = 3y^3 - 18y^2 + 36y - 24 - 4y^2 + 16y - 16 + 5y - 12 = 3y^3 - 22y^2 + 57y - 52.

📌 Thus, $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $. Expanding:

📌 3(x^6 - 6x^4 + 12x^2 - 8) - 22(x^4 - 4x^2 + 4) + 57x^2 - 114 - 52 = 3x^6 - 18x^4 + 36x^2 - 24 - 22x^4 + 88x^2 - 88 + 57x^2 - 166 = 3x^6 - 40x^4 + 181x^2 - 278.

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