We have the following expression:
[tex]\frac{a^2-5a+4}{3a+6}\cdot\frac{2a+4}{a^2-16}\text{.}[/tex]In order to do the multiplication, we simply multiply the denominators and put the result in the denominator, and we multiply the denominators and put then in the denominator:
[tex]\frac{(a^2-5a+4)(2a+4)}{(3a+6)(a^2-16)}\text{.}[/tex]Before we continue, we need to determine the values of the variables for which the expression is not defined. Since we have a division, this will only happen if the denominator is equal to 0.
Since we have a multiplication in the denominator, the only way it will be 0 is if either of the factors is 0. In other words, we can't have
[tex]3a+6=0,[/tex]or
[tex]a^2-16=0.[/tex]For the first equation, we simply subtract 6 from both sides and hen divide by 3 to obtain
[tex]a=-\frac{6}{3}=-2,[/tex]so a cannot be -2.
For the second equation, we can actually factor is further like this:
[tex]a^2-16=(a+4)(a-4),[/tex]and from this we can see that it will be 0 if a is either 4 o -4. In other words, and putting all of this together, a cannot be -2, -4 or 4.
With that out of the way, let's proceed with the multiplication:
[tex]\frac{(a^2-5a+4)(2a+4)}{(3a+6)(a^2-16)}=\frac{2a^3+4a^2-10a^2-20a+8a+16}{3a^3-48a+6a^2-96}\text{.}[/tex]Grouping similar terms together gives us:
[tex]\frac{2a^3+4a^2-10a^2-20a+8a+16}{3a^3-48a+6a^2-96}=\frac{2a^3-6a^2-12a+16}{3a^3+6a^2-48a-96}\text{.}[/tex]