Question:

Step 1: Understand the problem and initial approach We need to find the primitive (antiderivative) of the given function $f(x) = \frac{3x^4 - 1}{(x^4 + x + 1)^2}$. Direct integration of this form can be complex. A common strategy for such problems, especially in multiple-choice questions, is to check the derivatives of the given options. The primitive $F(x)$ must satisfy $F'(x) = f(x)$. Step 2: Calculate the derivative of Option 2 Let's consider Option 2, $F_2(x) = -\frac{x}{x^4 + x + 1}$. We use the quotient rule for differentiation, $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. Here, $u = x$ and $v = x^4 + x + 1$. Then $u' = 1$ and $v' = 4x^3 + 1$. $$F_2'(x) = -\frac{(1)(x^4 + x + 1) - (x)(4x^3 + 1)}{(x^4 + x + 1)^2}$$ $$F_2'(x) = -\frac{x^4 + x + 1 - 4x^4 - x}{(x^4 + x + 1)^2}$$ $$F_2'(x) = -\frac{-3x^4 + 1}{(x^4 + x + 1)^2}$$ $$F_2'(x) = \frac{3x^4 - 1}{(x^4 + x + 1)^2}$$ Step 3: Compare the derivative of Option 2 with the integrand The calculated derivative $F_2'(x) = \frac{3x^4 - 1}{(x^4 + x + 1)^2}$ exactly matches the given integrand $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$. This indicates that Option 2 is a primitive of the given function. Step 4: Calculate the derivative of Option 4 The problem statement indicates that Option D is the correct answer. Let's calculate the derivative of Option 4, $F_4(x) = -\frac{x+1}{x^4 + x + 1}$, to verify this. Using the quotient rule with $u = x+1$ and $v = x^4 + x + 1$. Then $u' = 1$ and $v' = 4x^3 + 1$. $$F_4'(x) = -\frac{(1)(x^4 + x + 1) - (x+1)(4x^3 + 1)}{(x^4 + x + 1)^2}$$ $$F_4'(x) = -\frac{x^4 + x + 1 - (4x^4 + x + 4x^3 + 1)}{(x^4 + x + 1)^2}$$ $$F_4'(x) = -\frac{x^4 + x + 1 - 4x^4 - x - 4x^3 - 1}{(x^4 + x + 1)^2}$$ $$F_4'(x) = -\frac{-3x^4 - 4x^3}{(x^4 + x + 1)^2}$$ $$F_4'(x) = \frac{3x^4 + 4x^3}{(x^4 + x + 1)^2}$$ Step 5: Compare the derivative of Option 4 with the integrand The calculated derivative $F_4'(x) = \frac{3x^4 + 4x^3}{(x^4 + x + 1)^2}$ does NOT match the given integrand $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$. Step 6: Conclusion Based on our calculations, the derivative of Option 2, $F_2(x) = -\frac{x}{x^4 + x + 1}$, precisely matches the given integrand. The derivative of Option 4 does not match the integrand. Therefore, Option 2 is the correct primitive. There might be an inconsistency between the question's stated correct answer and the actual mathematical derivation. The primitive of $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$ is $-\frac{x}{x^4 + x + 1} + c$. This corresponds to Option 2.