Evaluate the expression \[ (a^2+b)^2 - (a^2-b)^2, \]if $a=4$ and $b=1$.
First, I replace $a$ with 4 and $b$ with 1 in the expression: \[ (4^2+1)^2 - (4^2-1)^2 \] Next, I simplify the exponents inside the parentheses: \[ (16+1)^2 - (16-1)^2 \] Then, I simplify the additions and subtractions inside the parentheses: \[ (17)^2 - (15)^2 \] Now, I evaluate the squares of 17 and 15: \[ 289 - 225 \] Finally, I subtract 225 from 289: \[ 64 \] 