C. \( (4 a-b)^{2}=16 a^{2}-8 a b-b^{2} \)

To solve the given equation \( (4a-b)^{2} = 16a^{2} - 8ab - b^{2} \), we can expand the left side of the equation using the formula for squaring a binomial. The formula for squaring a binomial \( (x+y)^{2} \) is \( x^{2} + 2xy + y^{2} \). In this case, \( x = 4a \) and \( y = -b \). So, expanding \( (4a-b)^{2} \) gives: \[ (4a-b)^{2} = (4a)^{2} + 2(4a)(-b) + (-b)^{2} \] \[ (4a-b)^{2} = 16a^{2} - 8ab + b^{2} \] Now, we can compare this expanded form with the given equation: \[ 16a^{2} - 8ab + b^{2} = 16a^{2} - 8ab - b^{2} \] By comparing the two sides, we can see that the equation is true for all values of \( a \) and \( b \).