To determine which graph could represent the function $f(x) = x^4 + x^3 - x^2 - x$, we need to analyze the function's general behavior, including its end behavior, critical points, and possible inflection points.

Step-by-Step Analysis:

1. Degree and Leading Coefficient:

   • The function $f(x)$ is a polynomial of degree 4, with the leading term $x^4$.
   • Since the leading coefficient of $x^4$ is positive, the graph will rise to $+\infty$ as $x$ approaches $\infty$ and also as $x$ approaches $-\infty$.

2. End Behavior:

   • As $x \to \infty$, $f(x) \to \infty$.
   • As $x \to -\infty$, $f(x) \to \infty$.

3. Critical Points:

   • To find the critical points, we take the derivative $f'(x)$ and set it to zero: $f'(x) = 4x^3 + 3x^2 - 2x - 1$ Solving $f'(x) = 0$ will give us the $x$-coordinates of the critical points. These points are where the slope is zero, which could be a local maximum, minimum, or a point of inflection.

4. Possible Turning Points:

   • The function may have up to 3 turning points because the derivative is a cubic polynomial.
   • The signs of the coefficients indicate that the graph will likely have a local minimum and maximum.

5. Behavior at the Origin:

   • The function value at $x = 0$ is $f(0) = 0$.
   • Since $f(x)$ is a degree 4 polynomial, it is symmetric around the origin but does not need to be even or odd.

Expected Shape of the Graph:

• The graph should start high on the left (approaching $+\infty$ as $x$ approaches $-\infty$), decrease towards a local minimum, rise again to a local maximum, and then increase to $+\infty$ as $x$ approaches $\infty$.
• The graph will cross the x-axis at points corresponding to the roots of the function.

Possible Graph Shapes:

1. The graph will not be a simple parabola or straight line.
2. It will have a "W" or "M" shape depending on the relative position of the local maxima and minima.

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