Find $a$ and $b$ with whom this expression $x\bullet y=(a+x)(b+y)$ is associative I need to find a and b with whom this expression is associative: $$x\bullet y=(a+x)(b+y)$$
Also known that $$x,y\in Z$$
So firstly I write: $$(x\bullet y)\bullet z=x\bullet (y\bullet z)$$
Then I express and express them and after small algebra I get: $$ab^2+b^2x+az+abz+bxz=a^2b+a^2z+bx+abx+axz$$
And I don't know what to do now, I don't know if I should do it, but I tried to assign to $x=0$ and $z=1$ but nothing better happened.
Could you please share some ideas?
EDIT: Sorry, but I forgot to mention that I need to find such a and b with whom it should be associative

Assuming that you still want the element $1 \in \mathbb{Z}$ to be a multiplicative identity element with your new multiplication, here is an alternative way of quickly seeing that $a=b=0$.
For an arbitrary $y\in \mathbb{Z}$ we shall have:
$y=1 \bullet y=(a+1)(b+y)=ab+ay+b+y=(a+1)b+(a+1)y$
This is satisfied if and only if $a+1=1$ and $(a+1)b=0$. The unique solution is $a=0$ and $b=0$.
