Question

\[y\text{''}[t]+2y'[t]+y[t]=0\]

Answer 1

I get the first term but not the second
\[y[t]=e^{-t} C[1]+e^{-t} t C[2]\]

Answer 2

Its because you end up with a double root when you try to solve:
\[r^2+2r+1 = 0\]
basically you solution is in the form:
\[y(t) = (c_1+c_2t)e^{-t}\]