Step 1 :First, let's check Rahim's solution by expanding it:
Step 2 :\(16x^2 + 4y^2 = 4(4x^2 + y^2)\)
Step 3 :\(16x^2 + 4y^2 = 16x^2 + 4y^2\)
Step 4 :Rahim's solution matches the original expression.
Step 5 :Next, let's check Kate's solution by expanding it:
Step 6 :\(16x^2 + 4y^2 = 4(2x + y)(2x - y)\)
Step 7 :\(16x^2 + 4y^2 = 16x^2 - 4y^2\)
Step 8 :Kate's solution does not match the original expression.
Step 9 :\(\boxed{\text{Rahim is correct, and Kate is incorrect.}}\)