Solution: Set $ 3x + 2 = -2x + 12 $. Solve: $ 5x = 10 $, so $ x = 2 $. Substitute into $ y = 3(2) + 2 = 8 $. The intersection is $ oxed{(2, 8)} $.Question: Let $ g(x) $ be a polynomial such that $ g(2x + 3) = 4x^2 + 12x + 5 $. Find $ g(x^2 - 1) $.

["SEO-Optimized Article: Understanding How to Find $ g(x^2 - 1) $ Given $ g(2x + 3) = 4x^2 + 12x + 5 $ – Step-by-Step Solution", "Introduction\nWorking with polynomial transformations like $ g(2x + 3) $ reveals powerful techniques for analyzing functions. When you’re given $ g(2x + 3) = 4x^2 + 12x + 5 $ and asked to find $ g(x^2 - 1) $, the key lies in simplifying the input transformation and strategically substituting values. This article walks you through solving $ g(x^2 - 1) $ step by step, delivering clear insight for students, educators, and math enthusiasts.", "---", "Step 1: Understand the Functional Relationship\nWe are told:\n$$\ng(2x + 3) = 4x^2 + 12x + 5\n$$\nOur goal is to express $ g(u) $ in terms of $ u $, then substitute $ u = x^2 - 1 $ to compute $ g(x^2 - 1) $.", "Let $ u = 2x + 3 $. This substitution enables us to rewrite the right-hand side in terms of $ u $.", "---", "Step 2: Solve for $ x $ in Terms of $ u $\nFrom $ u = 2x + 3 $, solve for $ x $:\n$$\nu = 2x + 3 \Rightarrow x = \frac{u - 3}{2}\n$$", "---", "Step 3: Substitute $ x = \frac{u - 3}{2} $ into the Given Equation\nNow substitute into $ g(u) = 4x^2 + 12x + 5 $:\n$$\ng(u) = 4\left( \frac{u - 3}{2} \right)^2 + 12\left( \frac{u - 3}{2} \right) + 5\n$$", "Simplify each term:\n$$\n\left( \frac{u - 3}{2} \right)^2 = \frac{(u - 3)^2}{4}\n\Rightarrow 4 \cdot \frac{(u - 3)^2}{4} = (u - 3)^2\n$$\n$$\n12 \cdot \frac{u - 3}{2} = 6(u - 3)\n$$\nSo:\n$$\ng(u) = (u - 3)^2 + 6(u - 3) + 5\n$$", "---", "Step 4: Expand and Simplify the Expression\nFirst, expand $ (u - 3)^2 = u^2 - 6u + 9 $\nThen:\n$$\ng(u) = u^2 - 6u + 9 + 6u - 18 + 5\n$$\nCombine like terms:\n$$\ng(u) = u^2 + (-6u + 6u) + (9 - 18 + 5) = u^2 - 4\n$$", "Thus,\n$$\ng(u) = u^2 - 4\n$$", "---", "Step 5: Compute $ g(x^2 - 1) $\nNow substitute $ u = x^2 - 1 $ into $ g(u) $:\n$$\ng(x^2 - 1) = (x^2 - 1)^2 - 4\n$$\nExpand:\n$$\n(x^2 - 1)^2 = x^4 - 2x^2 + 1\n\Rightarrow g(x^2 - 1) = x^4 - 2x^2 + 1 - 4 = x^4 - 2x^2 - 3\n$$", "---", "Conclusion\nBy carefully transforming variables and simplifying the polynomial expression, we determined:\n$$\n\boxed{g(x^2 - 1) = x^4 - 2x^2 - 3}\n$$\nThis elegant solution demonstrates how function composition and substitution unlock deeper algebraic structure—essential for advanced problem solving in algebra.", "---", "Keywords: polynomial function $ g(x) $, $ g(2x + 3) = 4x^2 + 12x + 5 $, solve for $ g(x^2 - 1) $, find $ g(x) $, algebraic substitution, function transformation, high school algebra, math problem solving."]