Given that `x,y,z` are positive reals such that `xyz=32` . The minimum value of `x^2+4xy+4y^2+2z^2` is ___________.

Explanation

To find the minimum value of the expression \( E = x^2 + 4xy + 4y^2 + 2z^2 \) given that \( xyz = 32 \) and \( x, y, z \) are positive reals, we can apply the AM-GM inequality.

### Step 1: Rewrite the expression
We can rewrite the expression \( E \) as:
\[ E = x^2 + 4xy + 4y^2 + 2z^2 = x^2 + 2xy + 2xy + 4y^2 + z^2 + z^2 \]
This allows us to group terms conveniently for applying AM-GM.

### Step 2: Apply AM-GM inequality
By the AM-GM inequality, we have:
\[ \frac{x^2 + 2xy + 2xy + 4y^2 + z^2 + z^2}{6} \geq \sqrt[6]{x^2 \cdot (2xy)^2 \cdot 4y^2 \cdot z^2 \cdot z^2} \]
Calculating the product:
\[ x^2 \cdot (2xy)^2 \cdot 4y^2 \cdot z^2 \cdot z^2 = x^2 \cdot 4x^2y^2 \cdot 4y^2 \cdot z^4 = 16x^4y^4z^4 \]

### Step 3: Substitute \( xyz = 32 \)
Since \( xyz = 32 \), we can express \( x^4y^4z^4 \) as:
\[ (xyz)^4 = 32^4 = 1048576 \]
Thus, we have:
\[ E \geq 6 \sqrt[6]{16 \cdot 1048576} \]

### Step 4: Simplify the expression
Calculating \( 16 \cdot 1048576 \):
\[ 16 \cdot 1048576 = 16777216 \]
Now, we need to find the sixth root:
\[ \sqrt[6]{16777216} = 16 \]
Thus, we have:
\[ E \geq 6 \cdot 16 = 96 \]

### Conclusion
The minimum value of \( E = x^2 + 4xy + 4y^2 + 2z^2 \) given \( xyz = 32 \) is:
\[ \boxed{96} \]