Factor the polynomial $(x - 5)^2 + 6(x - 5) + 8$ using substitution.

Step 1: The binomial $(x - 5)$ appears in the middle term and is squared in the first term. Let $u = x - 5$. Step 2: Substitute $u$ to rewrite the expression as a quadratic trinomial: $u^2 + 6u + 8$ Step 3: Factor the trinomial. Find two numbers that multiply to $8$ and add to $6$. These are $2$ and $4$: $(u + 2)(u + 4)$ Step 4: Replace $u$ with the original binomial $(x - 5)$: $((x - 5) + 2)((x - 5) + 4)$ Step 5: Simplify the contents of the parentheses: $(x - 3)(x - 1)$

The factored form is $(x - 3)(x - 1)$.