Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I got stuck and need a bit of help. Here is what I have so far:

Try out some cases:

Let $y=0$: $$\begin{align} f(x)&=f(x)+f(0)+f(x)f(0) \\ 0&=f(0)+f(x)f(0) \\0 & = f(0)[1+f(x)] \end{align}$$
Observe that either $f(0)=0$ or $f(x)=-1$. So this gives me one solution, but I am having trouble finding the other solution(s). Somebody suggested to me that $f(x)=0$ is also a solution but I can't find a way to prove what they said is true. Can anyone please, without giving away the answer, give me a teeny hint? I really want to figure this out as much as I can. I've tried the case when $y=-x$ and $x=y$ but I don't feel like those cases help me towards the solution.

Thanks in advance