If $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = x^2 - 3x + 2$, find $f(f(x))$.

Step 1: Set up the composite function

To evaluate the composite function $f(f(x))$, replace every occurrence of the input variable $x$ in $f(x) = x^2 - 3x + 2$ with the expression for $f(x)$ itself, which is $x^2 - 3x + 2$.

Step 2: Expand the squared trinomial

Apply the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ with $a = x^2$, $b = -3x$, and $c = 2$. Combining the like terms gives $x^4 - 6x^3 + 13x^2 - 12x + 4$.

Step 3: Expand the linear term and constant

Distribute the factor of $-3$ across the terms inside $(x^2 - 3x + 2)$ to get $-3x^2 + 9x - 6$, and then add the constant $+2$, which simplifies to $-3x^2 + 9x - 4$.

Step 4: Combine all terms to find final expression

Add the two expanded parts together by combining like powers of $x$: $x^4$ remains as is; $-6x^3$ remains as is; for the degree 2 term, $13x^2 - 3x^2 = 10x^2$; for the degree 1 term, $-12x + 9x = -3x$; and for the constants, $4 - 4 = 0$.

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