Let us write the two complex numbers in polar coordinates and let them be
z_1=r_1(cosalpha+isinalpha) and z_2=r_2(cosbeta+isinbeta)
Here, if two complex numbers are a_1+ib_1 and a_2+ib_2 r_1=sqrt(a_1^2+b_1^2), r_2=sqrt(a_2^2+b_2^2) and alpha=tan^(-1)(b_1/a_1), beta=tan^(-1)(b_2/a_2)
Their multiplication leads us to
{r_1r_2}{(cosalpha+isinalpha)(cosbeta+isinbeta)} or
(r_1r_2){(cosalphacosbeta+i^2sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta)) or
(r_1r_2){(cosalphacosbeta-sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta)) or
(r_1r_2)*(cos(alpha+beta)+isin(alpha+beta)) or
z_1*z_2 is given by (r_1r_2, (alpha+beta))
So for multiplication complex number z_1 by z_2 , take new angle as (alpha+beta) and modulus is r_1*r_2 i.e. product of the modulus of two numbers.
Here 4+3i can be written as r_1(cosalpha+isinalpha) where r_1=sqrt(4^2+3^2)=sqrt25=5 and alpha=tan^(-1)3/4
and -3+2i can be written as r_2(cosbeta+isinbeta) where r_2=sqrt((-3)^2+2^2)=sqrt13 and beta=tan^(-1)(-2/3)
and z_1*z_2=5sqrt13(costheta+isintheta), where theta=alpha+beta
Hence, tantheta=tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)=(3/4+(-2/3))/(1-3/4xx(-2/3))=(1/12)/(3/2)=1/18.
Hence, (4+3i)*(-3+2i)=5sqrt13(costheta+isintheta), where theta=tan^(-1)(1/18)
