This is a simple Riccati equation:
now it is easy to see that y= 4 and y = -4/9 are particular solutions to this equation, the following substitution allows us to find the general solution:
which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is:
thus we get:
which is a linear first order ODE, and can be easily solved using the integrating factor method. After you finish just back substitute twice to obtain the final solution...