de Rham cohomology groups with compact support

Question

I am trying to solve the following questions regarding the de Rham cohomology groups with compact support: 1) $H^*_c(X):=\{H_c^k(X)\}^\infty_k$ where $k\geq 0$ is a graded algebra of $H^*(X)$ $\quad$ 2) if $X$ is a compact smooth manifold then $\Omega_c(X)=\Omega(X)$ and therefore $H^*_c(X)=H^*(X)$. Note: $\Omega_c(X)$ is the set of all smooth $k$-forms on a manifold $X$ with a compact support.