If \( a^{2}+b^{2}=z \) and \( a b=y \), which of the following is equivalent to \( 4 z+8 y \) ? A) \( (a+2 b)^{2} \) B) \( (2 a+2 b)^{2} \) C) \( (4 a+4 b)^{2} \) D) \( (4 a+8 b)^{2} \)

To find an expression equivalent to \( 4z + 8y \) given \( a^{2} + b^{2} = z \) and \( ab = y \), we start by substituting the values of \( z \) and \( y \). First, substitute \( z \) and \( y \): \[ 4z + 8y = 4(a^{2} + b^{2}) + 8(ab) \] Now, factor this expression: \[ = 4a^{2} + 4b^{2} + 8ab \] We can recognize that \( 4a^{2} + 8ab + 4b^{2} \) can be rewritten as a perfect square: \[ = (2a + 2b)^{2} \] Now we check whether this corresponds to one of the answer choices: A) \( (a + 2b)^{2} \) B) \( (2a + 2b)^{2} \) C) \( (4a + 4b)^{2} \) D) \( (4a + 8b)^{2} \) We see that our derived expression matches choice **B**: \[ (2a + 2b)^{2} \] Thus, the equivalent expression to \( 4z + 8y \) is \( (2a + 2b)^{2} \). The answer is B.