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i) $\mathrm{Apply\:binomial\:theorem}:\quad \left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i$

$\sum _{i=0}^6\binom{6}{i}\cdot \:2^{\left(6-i\right)}\left(3x\right)^i$

*After Plugging in Calc*

$\frac{6!}{0!\left(6-0\right)!}\cdot \:2^6\left(3x\right)^0+\frac{6!}{1!\left(6-1\right)!}\cdot \:2^5\left(3x\right)^1+\frac{6!}{2!\left(6-2\right)!}\cdot \:2^4\left(3x\right)^2+\frac{6!}{3!\left(6-3\right)!}\cdot \:2^3\left(3x\right)^3+\frac{6!}{4!\left(6-4\right)!}\cdot \:2^2\left(3x\right)^4+\frac{6!}{5!\left(6-5\right)!}\cdot \:2^1\left(3x\right)^5+\frac{6!}{6!\left(6-6\right)!}\cdot \:2^0\left(3x\right)^6$

$64+576x+2160x^2+4320x^3+4860x^4+2916x^5+729x^6$