The two terms are x and 2, thus, x+2 is a binomial. We have to multiply the binomial by itself four times since it is raised to 4th power.
Let us multiply x+2 by itself using Polynomial Multiplication:
(x+2)(x+2) = x^2 + 4x + 4
Taking the result, let us multiply it again by a+b:
(x^2 + 4x + 4)(x+2) = x^3 + 6x^2 + 12x + 8
And again:
(x^3 + 6x^2 + 12x + 8)(x+2) = x^4 + 8x^3 + 24x^2 + 32x +16
The binomial expansion of (x+2)^4 is x^4 + 8x^3 + 24x^2 + 32x +16