Let $f,g \in H^{1}(a,b)$ with $(f +g) \in H^{2}(a,b)$ where $0 < a < b < \infty$ and $$ f = ( g^{\prime} + f^{\prime} )^{\prime} \quad \text{in} \quad L^{2}(a,b) $$ then $$ \|f\|_{H^{-1}(a,b)} \leq C \|g^{\prime}\|_{L^{2}(a,b)} + C\|f^{\prime}\|_{L^{2}(a,b)} $$
I think that, since $L^{2}(a,b) \subset H^{-1}(a,b)$ with continuous injection, then there is $C > 0$ such that $$ \|f\|_{H^{-1}(a,b)} \leq C \|f\|_{L^{2}(a,b)} = C\|( g^{\prime} + f^{\prime} )^{\prime} \|_{L^{2}(a,b)} \leq C \| g^{\prime} + f^{\prime} \|_{H^{1}(a,b)} $$
But, I can't conclude.