x² + 16x + a = (x + b)²

 x² + 16x + a = (x + b)(x + b)

 x² + 16x + a = x² + bx + bx + b²

 x² + 16x + a = x² + 2bx + b²

Since the left and right sides must be
equal for all values of x, the corresponding
terms on each side must be equal for all x.

So we have 

 x² = x²
16x = 2bx
  a = b²

The second equation tells us 16 = 2b or b = 8

Substituting b = 8 into the third equation
gives  
 
a = b²
a = 8²
a = 64

Edwin