1. Problem statement: We are given functions $f,g:\mathbb{R}\to\mathbb{R}$ with $f(x)=3x+1$ and $(f+g)(x)=x^3+2x-1$. 2. Formula and key rule: To find $g(x)$ use the formula $g(x)=(f+g)(x)-f(x)$. 3. Substitute the given expressions: $$g(x)=x^3+2x-1-(3x+1)$$ 4. Simplify step by step. 4.1. Expand and combine like terms: $$g(x)=x^3+2x-1-3x-1$$ 4.2. Combine the $x$ terms and constants: $$g(x)=x^3 - x -2$$ 5. Evaluate at $x=-1$. 5.1. Substitute $x=-1$: $$g(-1)=(-1)^3 -(-1) -2$$ 5.2. Compute the arithmetic: $$g(-1)=-1+1-2$$ 5.3. Final value: $$g(-1)=-2$$ 6. Conclusion: The value of $g(-1)$ is $-2$ and the correct choice is (a) -2.