If $f(x) = x - 1$, and $g(f(x)) = 3 + 2x - x^2 + x^3$, which of the options below give a correct expression for $g(x)$?

Since $f(x) = x - 1$, the given relation is:

$g(x - 1) = x^3 - x^2 + 2x + 3$

To get $g$ in terms of its own input, let $t = x - 1$, so $x = t + 1$.

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$g(t) = (t + 1)^3 - (t + 1)^2 + 2(t + 1) + 3$

Using $(t + 1)^3 = t^3 + 3t^2 + 3t + 1$ and $(t + 1)^2 = t^2 + 2t + 1$:

$g(t) = t^3 + 3t^2 + 3t + 1 - t^2 - 2t - 1 + 2t + 2 + 3$

$= t^3 + 2t^2 + 3t + 5$

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$t$ is just a name for the input, so replacing $t$ with $x$:

$g(x) = x^3 + 2x^2 + 3x + 5$

Quick shortcut: at $x = 1$, $f(1) = 0$, so $g(0) = 3 + 2 - 1 + 1 = 5$. Only $x^3 + 2x^2 + 3x + 5$ gives $5$ at $x = 0$.