I know that the degree of the quotient is the difference between the degrees of the dividend and the divisor, so $\deg q = \deg f - \deg d$. I also know that the degree of the remainder is always less than the degree of the divisor, so $\deg r < \deg d$. I want to maximize $\deg q$, so I want to minimize $\deg d$. The smallest possible value of $\deg d$ is one more than the degree of the remainder, since $\deg r < \deg d$. So the smallest possible value of $\deg d$ is $4$, and the maximum possible value of $\deg q$ is $9 - 4 = 5$. 
The polynomial $f(x)$ is divided by the polynomial $d(x)$ to give a quotient of $q(x)$ and a remainder of $r(x)$. If $\deg f = 9$ and $\deg r = 3$, what is the maximum possible value of $\deg q$?