The rule for deriving a multiplication is
[tex] (f(x)\cdot g(x))' = f'(x)g(x)+f(x)g'(x) [/tex]
Also, we can forget about the factor 4 for now, since we have
[tex] (4f(x)\cdot g(x))' = 4(f'(x)g(x)+f(x)g'(x)) [/tex]
So, we will just multiply everything by 4 at the end. Our functions are
[tex] f(x) = (x^7-9)^{12},\quad g(x) = (4x+8)^{11} [/tex]
For both derivatives we will use the rule
[tex] (f(x)^n)' = n\cdot f(x)^{n-1}\cdot f'(x) [/tex]
So, we have
[tex] f'(x) = 12\cdot(x^7-9)^{11}\cdot 7x^6,\quad g'(x) = 11\cdot(4x+8)^{10}\cdot 4 [/tex]
We can simplify those expression a little bit:
[tex] f'(x) = 84x^6(x^7-9)^{11},\quad g'(x) = 44(4x+8)^{10}[/tex]
The formula [tex] f'(x)g(x)+f(x)g'(x) [/tex] thus becomes
[tex] 84x^6(x^7-9)^{11}\cdot (4x+8)^{11} + (x^7-9)^{12} \cdot 44(4x+8)^{10} [/tex]
And so the final answer is
[tex]4(84x^6(x^7-9)^{11}\cdot (4x+8)^{11} + (x^7-9)^{12} \cdot 44(4x+8)^{10})[/tex]
If you simplify this expression by factoring common terms, you will see that the correct answer is the first one.
