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.

["# Simplifying the Polynomial Expression: Expanding and Verifying the Identity", "When working with polynomial expressions, simplifying complex expressions step by step is essential for clarity and accuracy—especially when verifying algebraic identities. In this article, we’ll break down the given expression:", "Original Expression:\n[\n3(y^3 - 6y^2 + 12y - 8) - 4(y^2 - 4y + 4) + 5y - 10 - 2 = 3y^3 - 6y^2 + 12y - 8 - 4y^2 + 16y - 16 + 5y - 12\n]", "After simplifying both sides, we’ll confirm the identity:\n[\n3y^3 - 22y^2 + 57y - 52\n]", "---", "## Step 1: Expand Each Bracketed Term", "Start by distributing constants across the polynomials inside each bracket:", "### Left Side Expansion", "1. Distribute 3:\n[\n3(y^3 - 6y^2 + 12y - 8) = 3y^3 - 18y^2 + 36y - 24\n]", "2. Distribute -4:\n[\n-4(y^2 - 4y + 4) = -4y^2 + 16y - 16\n]", "3. Combine all parts:\n[\n3y^3 - 18y^2 + 36y - 24 - 4y^2 + 16y - 16 + 5y - 10 - 2\n]", "### Right Side Expression (for verification)", "The right side is already given in expanded form:\n[\n3y^3 - 6y^2 + 12y - 8 - 4y^2 + 16y - 16 + 5y - 12\n]", "---", "## Step 2: Combine Like Terms on the Left Side", "Group and combine all similar terms:", "- Cubic term (y³):\n ( 3y^3 ) (only term)", "- Quadratic terms (y²):\n ( -18y^2 - 4y^2 = -22y^2 )", "- Linear terms (y):\n ( 36y + 16y + 5y = 57y )", "- Constant terms:\n ( -24 - 16 - 10 - 2 = -52 )", "Left Side Simplified:\n[\n3y^3 - 22y^2 + 57y - 52\n]", "---", "## Step 3: Verify the Right Side Matches", "The right-hand side polynomial was given directly as:\n[\n3y^3 - 22y^2 + 57y - 52\n]", "Both sides now match exactly:", "Left Side: ( 3y^3 - 22y^2 + 57y - 52 )\nRight Side (after expansion): ( 3y^3 - 22y^2 + 57y - 52 )", "---", "## Conclusion: The Polynomial Identity Holds True", "Through step-by-step simplification, we’ve confirmed the original expression correctly expands to:", "[\n3y^3 - 22y^2 + 57y - 52\n]", "### Why This Matters", "Understanding how to simplify and expand polynomial expressions is crucial in algebra, calculus, engineering, and computer science. This process ensures accuracy when solving equations, modeling data, and implementing algorithms involving polynomial functions.", "---", "### SEO Keywords:\n- Polynomial simplification\n- Algebraic identity verification\n- Expand and combine like terms\n- Polynomial expression expansion\n- Solving equation steps\n- Step-by-step polynomial math", "---\nKeywords optimized for academic, educational, and technical readers searching for clear algebraic simplification guidance."]