1. The problem is to factor the quadratic expression $r^2 - 12r + 35$ into the form $(r - a)(r - b)$. 2. We use the factoring method for quadratics: find two numbers $a$ and $b$ such that $a \times b = 35$ (the constant term) and $a + b = 12$ (the coefficient of $r$ with opposite sign because of subtraction). 3. The pairs of factors of 35 are $(1, 35)$ and $(5, 7)$. Among these, $5 + 7 = 12$, which matches the middle term coefficient. 4. Therefore, the factors are $5$ and $7$. Since the middle term is $-12r$, both factors are subtracted: $(r - 5)(r - 7)$. 5. To verify, expand: $$ (r - 5)(r - 7) = r^2 - 7r - 5r + 35 = r^2 - 12r + 35 $$ which matches the original expression. 6. Final answer: $\boxed{(r - 5)(r - 7)}$