1. **Stating the problem:** We are given two functions: $$f(x) = x^4 - 4x^3 + x^2 + 1$$ $$g(x) = 1 - 2x$$ and the composition: $$f(g(x)) = x^4 - 3x^3 + 2x^2$$ We want to verify the composition and understand how $f(g(x))$ is formed. 2. **Recall the composition formula:** The composition of functions $f$ and $g$ is defined as: $$ (f \circ g)(x) = f(g(x)) $$ This means we substitute $g(x)$ into every $x$ in $f(x)$. 3. **Substitute $g(x)$ into $f(x)$:** Given $g(x) = 1 - 2x$, substitute into $f(x)$: $$ f(g(x)) = (1 - 2x)^4 - 4(1 - 2x)^3 + (1 - 2x)^2 + 1 $$ 4. **Expand each term:** - Expand $(1 - 2x)^4$ using binomial theorem: $$ (1 - 2x)^4 = 1 - 8x + 24x^2 - 32x^3 + 16x^4 $$ - Expand $-4(1 - 2x)^3$: $$ (1 - 2x)^3 = 1 - 6x + 12x^2 - 8x^3 $$ Multiply by $-4$: $$ -4 + 24x - 48x^2 + 32x^3 $$ - Expand $(1 - 2x)^2$: $$ 1 - 4x + 4x^2 $$ 5. **Combine all terms:** $$ f(g(x)) = (1 - 8x + 24x^2 - 32x^3 + 16x^4) + (-4 + 24x - 48x^2 + 32x^3) + (1 - 4x + 4x^2) + 1 $$ 6. **Simplify by combining like terms:** - Constant terms: $1 - 4 + 1 + 1 = -1$ - $x$ terms: $-8x + 24x - 4x = 12x$ - $x^2$ terms: $24x^2 - 48x^2 + 4x^2 = -20x^2$ - $x^3$ terms: $-32x^3 + 32x^3 = 0$ - $x^4$ term: $16x^4$ So, $$ f(g(x)) = 16x^4 + 12x - 20x^2 - 1 $$ 7. **Compare with given $f(g(x)) = x^4 - 3x^3 + 2x^2$:** Our expansion does not match the given $f(g(x))$. This suggests either a different $g(x)$ or a different $f(x)$ was used in the problem statement. **Final answer:** The composition $f(g(x))$ based on the given $f$ and $g$ is: $$ f(g(x)) = 16x^4 + 12x - 20x^2 - 1 $$