The Binomial Expansion Theorem for a binomial elevated to the fourth power is: [tex](x+y) ^{4} = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4x y^{3} + y^{4} [/tex]
Now lets apply this to our expression:
[tex](y^{3} +3x)^{4} = y^{12} +12x y^{9}+54 x^{2} y^{6}+108 x^{3} y^{3} +81 x^{4} [/tex]
The third term in our binomial expansion is [tex]54 x^{2} y^{6} [/tex]; therefore, the correct answer is the fourth one.
