a \( E(x)=(x+2)^{3}-4 x-8 \)

To analyze the function \( E(x) = (x + 2)^3 - 4x - 8 \), we can start by expanding the expression and simplifying it. 1. **Expand \( (x + 2)^3 \)**: \[ (x + 2)^3 = x^3 + 3 \cdot 2x^2 + 3 \cdot 2^2 x + 2^3 = x^3 + 6x^2 + 12x + 8 \] 2. **Substitute back into \( E(x) \)**: \[ E(x) = (x^3 + 6x^2 + 12x + 8) - 4x - 8 \] 3. **Combine like terms**: \[ E(x) = x^3 + 6x^2 + (12x - 4x) + (8 - 8) \] \[ E(x) = x^3 + 6x^2 + 8x \] So, the simplified form of the function is: \[ E(x) = x^3 + 6x^2 + 8x \] ### Further Analysis If you want to analyze the function further, you can find its critical points, inflection points, or evaluate its behavior at certain values of \( x \). 1. **Finding Critical Points**: To find the critical points, we can take the derivative \( E'(x) \) and set it to zero: \[ E'(x) = 3x^2 + 12x + 8 \] Setting the derivative equal to zero: \[ 3x^2 + 12x + 8 = 0 \] You can use the quadratic formula to solve for \( x \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3} \] \[ = \frac{-12 \pm \sqrt{144 - 96}}{6} = \frac{-12 \pm \sqrt{48}}{6} = \frac{-12 \pm 4\sqrt{3}}{6} = \frac{-2 \pm \frac{2\sqrt{3}}{3}}{1} \] \[ = -2 \pm \frac{2\sqrt{3}}{3} \] 2. **Finding the Roots**: To find the roots of \( E(x) \), you can set \( E(x) = 0 \) and solve for \( x \): \[ x^3 + 6x^2 + 8x = 0 \] Factor out \( x \): \[ x(x^2 + 6x + 8) = 0 \] This gives one root at \( x = 0 \). The quadratic can be solved using the quadratic formula: \[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} = \frac{-6 \pm \sqrt{36 - 32}}{2} = \frac{-6 \pm 2}{2} \] \[ = -3 \pm 1 \Rightarrow x = -2 \text{ or } x = -4 \] Thus, the roots of \( E(x) \) are \( x = 0, -2, -4 \). If you have any specific questions or further analysis you would like to perform on this function, feel free to ask!