Normal function of euclidian integers

This is to show that the normal function of euclidian integers preserves the multiplicative property. To start lets define the euclidian integers:

$\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\}$ Then the normal function is defined as $N(a+bi)=(a+bi)(a-bi)=a^2+b^2$ Now to prove the corresponding homomorphism preserves multiplication: $N(a+bi)N(c+di)=(a^2+b^2)(c^2+d^2)=(a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2)$ And from the other side $N((a+bi)(c+di))=N(ac+adi+cbi-db)=N((ac-dv)+(ad+cb)i)=(ac-dv{)}^2+(ad+cb{)}^2$