#### Answer

$(q-6)(q-2)$

#### Work Step by Step

To factor a trinomial in the form $x^2+bx+c$, we must find two numbers whose product is $c$ and whose sum is $b$. We then insert these two numbers into the blanks of the factors $(x+\_)(x+\_)$.
In the case of $q^2-8q+12$, we are looking for two numbers whose product is $12$ and whose sum is $-8$. The numbers $-6$ and $-2$ meet these criteria, because $$-6\times(-2)=12\;\text{and}\;-6+(-2)=-8$$When we insert these numbers into the blanks, we arrive at the factors $(q-6)(q-2)$.
To check our answer, we use the FOIL method to multiply these two factors together. To use the FOIL method, we multiply the first term from each binomial, the first term from the first binomial and the last term from the second binomial, the last term of the first binomial and the first term of the second binomial, and the last terms of each binomial. We then find the sum of these products and combine the like terms to get our answer.
When we apply this method to $(q-6)(q-2)$, we get the answer $q^2-8q+12$. Therefore, our solution checks.