I want to make sure my reasoning is correct: Recall first that a bilinear form $a : V \times V \rightarrow \mathbb{R}$ on a normed space $V$ is coercive if $$a(u,u) \geq c\|u\|_V^2,$$for some $c > 0$.
I have the following bilinear form $a(u,v) = \int_\Omega \nabla u \cdot \nabla v dx$ on $H_0^1(\Omega)$. I want to show $a$ is coercive: Notice that $$a(u,u) = \int_\Omega |\nabla u|^2dx = \| \nabla u\|_2^2.$$Now, since $\Omega$ is bounded we have the Poincaré inequality that tell us that the norm $\| \nabla u\|_2$ is equivalent to the norm $\|u\|_{H^1}$, hence on $H^1_0$. So there's some constant $c > 0$ such that $$\|\nabla u\|_2^2 \geq c\|u\|^2_{H^10}.$$So we have $$a(u,u) \geq c\|u\|_{H_0^1}^2.$$
If we now consider the bilinear form $$b(u,v) = \int_\Omega \nabla u \cdot \nabla v + uvdx,$$then $$b(u,u) = \|\nabla u\|_2^2 + \|u\|_2^2 \geq \|u\|_{H^1_0}.$$
This shows that both bilinear forms are coercive.