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In $\mathbb{Z}_{7}[x]: f(x)=6(2x+3)^{3}(3x+5)^{5}(5x+1)^{7},\ g(x)=5(2x+3)^{6}(x+4)^{2}(4x+2)^{4}$. I've done a couple of problems with finding gcd but haven't dealt with one that has quantities to a power so my main question is do I just foil this out? I feel like there is a better way. Sorry if this seems like a trivial question I struggle a lot with abstract algebra

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Those coefficients in each factor are ugly and don't matter to the GCD. What do you get when you pull them all out? For example, $5x+1=5(x+?)$ in $\mathbb{Z}_7$? Once you do that, all that's left to do is to match the factors in $f$ and $g$ that are equal, and take the lesser power of each, and multiply them all together. For example, the way it is now, you can see that both have $3$ factors of $(2x+3)$. So in the GCD, you will have a factor of $(2x+3)^3=2^3(x+5)^3$. And remember, the $2^3$ in front doesn't matter.

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    A little confused here so I've factored out like you said and I have left the following $f(x)=6(2^{3})(x+5)^{3}(3^{5})(x+4)^{5}(5^{7})(x+3)^{7}$ and $g(x)= 5(2^{6})(x+5)^{6}(x+4)^{2}(4^{4})(x+4)^{4}$ so now i take the matching ones like $(x+5) $ and $ (x+4)$ and multiply them together? and that's the gcd? – 2017-02-19
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    Yes, it's that easy! This product is obviously a common divisor, and it's also obviously the greatest possible common divisor. Because what bigger polynomial could divide $f$ and $g$ both? – 2017-02-20