Factor the polynomial $(x - 2)^2 + 7(x - 2) + 12$ using the substitution method.

Step 1: Identify the repeated binomial expression. The binomial $(x - 2)$ in the middle term is squared in the first term. Let $u = x - 2$. Step 2: Substitute $u$ into the expression to rewrite it as a standard quadratic trinomial: $u^2 + 7u + 12$ Step 3: Factor the trinomial in terms of $u$. Find two numbers that multiply to $12$ and add to $7$. These numbers are $3$ and $4$: $(u + 3)(u + 4)$ Step 4: Replace $u$ with the original binomial expression, $(x - 2)$: $((x - 2) + 3)((x - 2) + 4)$ Step 5: Simplify the expressions inside the parentheses: $(x + 1)(x + 2)$

The completely factored form is $(x + 1)(x + 2)$. This demonstrates how substitution can be used when the replaced term is a binomial rather than a monomial.