To solve the equation \(-2x^2 + 4x + 4 = -2x + 8\), we first rearrange all terms to one side of the equation:
\[ -2x^2 + 4x + 4 + 2x - 8 = 0 \]
Combine the like terms:
\[ -2x^2 + (4x + 2x) + (4 - 8) = 0 \]
This simplifies to:
\[ -2x^2 + 6x - 4 = 0 \]
Now, we can multiply the entire equation by -1 to make the leading coefficient positive:
\[ 2x^2 - 6x + 4 = 0 \]
Next, we can use the quadratic formula, where \(a = 2\), \(b = -6\), and \(c = 4\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant:
\[ b^2 - 4ac = (-6)^2 - 4(2)(4) = 36 - 32 = 4 \]
Now substituting back into the quadratic formula:
\[ x = \frac{6 \pm \sqrt{4}}{2 \cdot 2} \]
This simplifies to:
\[ x = \frac{6 \pm 2}{4} \]
Now we can calculate the two possible solutions:
- \(x = \frac{6 + 2}{4} = \frac{8}{4} = 2\)
- \(x = \frac{6 - 2}{4} = \frac{4}{4} = 1\)
Thus, the two solutions to the equation are \(x = 2\) and \(x = 1\).
The larger solution is:
\[ \boxed{2} \]