By the binomial theorem,
[tex](2x+y)^4=\displaystyle\sum_{k=0}^4\binom 4k(2x)^{4-k}y^k=\sum_{k=0}^4\binom 4k2^{4-k}x^{4-k}y^k[/tex]
where
[tex]\dbinom nk=\dfrac{n!}{k!(n-k)!}[/tex]
Then the coefficients of the [tex]x^{4-k}y^k[/tex] terms in the expansion are, in order from [tex]k=0[/tex] to [tex]k=4[/tex],
[tex]\dbinom 402^{4-0}=1\cdot2^4=16[/tex]
[tex]\dbinom412^{4-1}=4\cdot2^3=32[/tex]
[tex]\dbinom422^{4-2}=6\cdot2^2=24[/tex]
[tex]\dbinom432^{4-3}=4\cdot2^1=8[/tex]
[tex]\dbinom442^{4-4}=1\cdot2^0=1[/tex]
