can be rewritten in the following way

this we can factorise as follows

so the equation has roots at -4 and 2 hence the answer to the homogenous DE is

where A and B are both constants

the second problem is almost the same, we have already determined the solution to the homogenous equation which had the roots mentioned above. Since the term on the right hand side does not correspondto any of the roots we can write the solution according to the superposition principle as follows

we can now also find C since we know the homogenous solution to yield zero we only need to do the same with the non homogenous solution. So plug into the differential equation and solve for C. You should find C=2, then you can write the general solution as follows

using the initial conditions you can then solve for A and B by just plugging them in.