The binomial theorem for a binomial in the format (x + y) to the fourth power is
x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
So now we plug in for the x- and y-values
a⁴ + 4a³(2b) + 6a²(2b)² + 4a(2b)³ + (2b)⁴
Simplify this to become:
a⁴ + 8a³b + 6a²(4b²) + 4a(8b³) + 16b⁴
Now simplify to:
a⁴ + 8a³b + 24a²b² + 32ab³ + 16b⁴