Proving an energy estimate of $2 \delta \|u\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}+\|b\|_{L^2(\Omega)} \|u_t\|_{L^2(\Omega)}$

Question

I want to prove that $$2 \delta \|u\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}+\|b\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}\leq \frac{1+2\sqrt{\delta}}{2}(\|u_t\|^2_{L^2(\Omega)}+\|u_x\|^2_{L^2(\Omega)}+\delta\|u\|^2_{L^2(\Omega)})+\frac{1}{2}\|b\|^2_{L^2(\Omega)}$$ and this is what I did for all $\epsilon>0$, $$ 2\delta \|u\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}\!+\|b\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}\!\leq \delta(\epsilon\|u\|^2_{L^2(\Omega)}+\frac{1}{\epsilon}\|u_t\|^2_{L^2(\Omega)})+\frac{1}{2}\|b\|^2_{L^2(\Omega)}+\frac{1}{2}\|u\|^2_{L^2(\Omega)}$$ but I could not get to the inequality. I did not find an adequate $\epsilon>0$, in order to have the first inequality.