To solve the equation \( 2x^2 + 8x = x^2 - 16 \), we first rearrange all terms to one side:
\[ 2x^2 + 8x - x^2 + 16 = 0 \]
Combining like terms gives:
\[ (2x^2 - x^2) + 8x + 16 = 0 \]
Simplifying this, we have:
\[ x^2 + 8x + 16 = 0 \]
Next, we can factor the quadratic equation:
\[ (x + 4)(x + 4) = 0 \]
This can also be written as:
\[ (x + 4)^2 = 0 \]
Setting the factor equal to zero, we find:
\[ x + 4 = 0 \]
Solving for \( x \) gives:
\[ x = -4 \]
Thus, the only solution to the equation \( 2x^2 + 8x = x^2 - 16 \) is
\[ \boxed{-4} \]