1. **State the problem:** Simplify the expression $8x^4 + 12x^2 + 18$ and verify the given steps. 2. **Analyze the original expression:** The expression is $8x^4 + 12x^2 + 18$. 3. **Check the first step given:** $2(2x^2) + 6(2x) + 18$ does not correctly represent the original expression because $2(2x^2) = 4x^2$ and $6(2x) = 12x$, which does not match the original terms. 4. **Simplify the original expression by factoring:** $$8x^4 + 12x^2 + 18 = 2(4x^4 + 6x^2 + 9)$$ 5. **Try to factor the quadratic in $x^2$:** Let $y = x^2$, then the expression inside parentheses is $4y^2 + 6y + 9$. 6. **Calculate the discriminant:** $$\Delta = 6^2 - 4 \times 4 \times 9 = 36 - 144 = -108 < 0$$ Since the discriminant is negative, the quadratic does not factor over the reals. 7. **Conclusion:** The expression $8x^4 + 12x^2 + 18$ cannot be simplified further by factoring over the real numbers. 8. **Regarding the other expressions:** - $2x^2 + 2x + 18$ is unrelated to the original expression. - $-x^2 - 2x$ is also unrelated. **Final answer:** The original expression $8x^4 + 12x^2 + 18$ is already in simplest form with no real factorization.