1. **Problem:** Find $k$ if $f(x) = 3x^4 - 2x^3 + kx + 7$ is divisible by $x+1$. 2. **Formula:** If $x+1$ is a factor, then $f(-1) = 0$ (Factor Theorem). 3. **Calculation:** $$f(-1) = 3(-1)^4 - 2(-1)^3 + k(-1) + 7 = 3 + 2 - k + 7 = 12 - k$$ Set equal to zero: $$12 - k = 0 \implies k = 12$$ 4. **Answer:** None of the options match $k=12$, so re-check the problem or options. Since the problem states divisibility by $x+1$, and the calculation shows $k=12$, but options are A. -8 B. 8 C. 4 D. -4, the closest is none. Possibly a typo in options. **Final answer:** $k=12$ (not listed). --- **Summary:** Using the Factor Theorem, substituting $x=-1$ into the polynomial and setting equal to zero gives $k=12$.