Where is the negative sign coming from in the distributive step I came across a proof in the Book Quantum Computer for Computer Scientists. It shows the derivation of the equation for the division of complex numbers. I am trying to follow. It starts with
$$
(x,y) = \frac{(a_{1}, b_{1})}{(a_{2}, b_{2})}
$$
where a1, b1 represent the real and imaginary part of a complex number c1. Likewise for a2 and b2. The next step in the proof says:

then by definition of division as the inverse of multiplication
$$
(a_{1}, b_{1}) = (x,y) \times (a_{2}, b_{2})
$$

which I understand fine, but then it says

or
$$
(a_{1}, b_{1}) = (a_{2}x - b_{2}y, a_{2}y + b_{2}x)
$$

My question is where does that (-) come from? everything is positive. I expect it to be:
$$
(a_{1}, b_{1}) = (a_{2}x + b_{2}y, a_{2}y + b_{2}x)
$$
I am guessing my understanding of how multiplication works here is wrong. Any help is much appreciated.

That negative sign comes from $i^2=-1$.
$(a_2+ib_2)(x+iy)=a_2x+ia_2y+ib_2x+\color{blue}{i^2}b_2y=(a_2x\color{blue}-b_2y)+i(a_2y+b_2x)$.
