calculating probability mass function Not quite sure how to approach this question - let's say you're looking at a toll booth lane that can hold up to $7$ cars at a time. 
$X$ is the $\#$ of cars in the lane at a randomly chosen time 
with the probability $X = x$ proportional to $(x + 1)(8 - x)$. 
How do you calculate the probability mass function (pmf) and then the probability that $X$ will be at least $5$? I have absolutely no idea how to calculate for the pmf, but I think calculating for $P(X > 5)$ goes like this:
\begin{array}{ll}
P(0) &= (0+1)(8-0) = (1)(8) = 8\\
P(1) &= (1+1)(8-1) = (2)(7) = 14\\
P(2) &= 18\\
P(3) &= 20\\
P(4) &= 20\\
P(5) &= 18\\
P(6) &= 14\\
P(7) &= 8\\
\end{array}
and so
$$
P(X > 5) = P(5)+P(6)+P(7) = 18 + 14 + 8 = 40.
$$
If someone could point me in the right direction, that'd be great.

You have overlooked the word proportional.
You don't have $P(0)=8$.  Instead, you know there is a constant $c$ such that $P(0)=8c$.  
Similarly, you have $P(1)=14c$, $P(2)=18c$, etc.
Now, use the fact that $P(0)+P(1)+P(2)+...+P(7)=1$ and solve for $c$.  Then you will be able to calculate all values of $P$ exactly.
