1. **State the problem:** Given the function $f(x) = 2x^2 + 4$ and the equation $f(x + 2) = 2x^2$, find the value of $x$. 2. **Use the function definition:** We know that $f(x + 2) = 2(x + 2)^2 + 4$ by substituting $x + 2$ into the function. 3. **Set up the equation:** According to the problem, $f(x + 2) = 2x^2$, so $$2(x + 2)^2 + 4 = 2x^2$$ 4. **Expand the square:** $$(x + 2)^2 = x^2 + 4x + 4$$ So, $$2(x^2 + 4x + 4) + 4 = 2x^2$$ 5. **Distribute and simplify:** $$2x^2 + 8x + 8 + 4 = 2x^2$$ $$2x^2 + 8x + 12 = 2x^2$$ 6. **Subtract $2x^2$ from both sides:** $$\cancel{2x^2} + 8x + 12 = \cancel{2x^2}$$ $$8x + 12 = 0$$ 7. **Solve for $x$:** $$8x = -12$$ $$x = \frac{-12}{8}$$ 8. **Simplify the fraction:** $$x = \frac{\cancel{-12}^{-3} }{\cancel{8}^2} = -\frac{3}{2}$$ **Final answer:** $$x = -\frac{3}{2}$$