What is the value of 16xy^2+32x^2y?

Great question!

Target question: What is the value of 32xy² + 16x²y?
After scanning the two statements, I see that we might benefit from rephrasing the target question.
32xy² + 16x²y = 16xy(2y + x)
So.....
REPHRASED target question: What is the value of 16xy(2y + x)?

Statement 1: (x+2y)² = 64
This tells us 2 things: EITHER x+2y = 8 OR x+2y = -8
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 0 and y = 8, in which case 16xy(2y + x) = 0
Case b: x = 1 and y = 7, in which case 16xy(2y + x) = some value other than 0
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x = 2y
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 0 and y = 0, in which case 16xy(2y + x) = 0
Case b: x = 2 and y = 1, in which case 16xy(2y + x) = some value other than 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that EITHER x+2y = 8 OR x+2y = -8
Statement 2 tells us that x = 2y

So, we have two possible systems of equations:
x+2y = 8
x = 2y

AND
x+2y = -8
x = 2y

If we solve the first system, we get: x = 4 and y = 2, in which case 16xy(2y + x) = 16(4)(2)[2(2) + (4)] = some POSITIVE number
If we solve the second system, we get: x = -4 and y = -2, in which case 16xy(2y + x) = 16(-4)(-2)[2(-2) + (-4)] = some NEGATIVE number
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E