The required coefficient is the coefficient of $x^{12}$ in $(1+x+2x^2)(x+1)^{10}$
Step-by-Step Solution
Key Concept: To find a specific coefficient in a product of polynomials, identify which terms multiply to give that power and sum their contributions.
Expand $(1+x+2x^2)(x+1)^{10}$. The coefficient of $x^{12}$ comes from: $1 \cdot ^{10}C_{12}$ (impossible), $x \cdot ^{10}C_{11}$, and $2x^2 \cdot ^{10}C_{10}$. This gives $^{10}C_{11} + 2 \cdot ^{10}C_{10} = 210 + 2(504) = 210 + 504 = 714$.
Correct Answer: 714