1. **Problem Statement:** Given the functions $f(x) = 2x^3 - 5x^2 + 4x - 1$ and $g(x) = x^2 - 3x + 2$, find the composite function $h(x) = f(g(x))$ and simplify it. 2. **Formula and Rules:** The composite function $h(x) = f(g(x))$ means we substitute $g(x)$ into every $x$ in $f(x)$. 3. **Step-by-step Solution:** - Start with $f(x) = 2x^3 - 5x^2 + 4x - 1$. - Substitute $x$ with $g(x) = x^2 - 3x + 2$: $$h(x) = 2(g(x))^3 - 5(g(x))^2 + 4(g(x)) - 1$$ - Calculate each term: - $(g(x))^3 = (x^2 - 3x + 2)^3$ - $(g(x))^2 = (x^2 - 3x + 2)^2$ 4. **Expand $(g(x))^2$:** $$(x^2 - 3x + 2)^2 = (x^2)^2 - 2 \cdot x^2 \cdot 3x + 2 \cdot x^2 \cdot 2 + (3x)^2 - 2 \cdot 3x \cdot 2 + 2^2$$ Simplify: $$= x^4 - 6x^3 + 4x^2 + 9x^2 - 12x + 4 = x^4 - 6x^3 + 13x^2 - 12x + 4$$ 5. **Expand $(g(x))^3$:** $$ (g(x))^3 = (g(x))^2 \cdot g(x) = (x^4 - 6x^3 + 13x^2 - 12x + 4)(x^2 - 3x + 2) $$ Multiply term by term and combine like terms: $$= x^6 - 9x^5 + 37x^4 - 78x^3 + 86x^2 - 48x + 8$$ 6. **Substitute back into $h(x)$:** $$h(x) = 2(x^6 - 9x^5 + 37x^4 - 78x^3 + 86x^2 - 48x + 8) - 5(x^4 - 6x^3 + 13x^2 - 12x + 4) + 4(x^2 - 3x + 2) - 1$$ 7. **Distribute coefficients:** $$= 2x^6 - 18x^5 + 74x^4 - 156x^3 + 172x^2 - 96x + 16 - 5x^4 + 30x^3 - 65x^2 + 60x - 20 + 4x^2 - 12x + 8 - 1$$ 8. **Combine like terms:** $$= 2x^6 - 18x^5 + (74x^4 - 5x^4) + (-156x^3 + 30x^3) + (172x^2 - 65x^2 + 4x^2) + (-96x + 60x - 12x) + (16 - 20 + 8 - 1)$$ $$= 2x^6 - 18x^5 + 69x^4 - 126x^3 + 111x^2 - 48x + 3$$ **Final answer:** $$h(x) = 2x^6 - 18x^5 + 69x^4 - 126x^3 + 111x^2 - 48x + 3$$