In trig form we have
(R_1 e^(i theta_1))/(R_2 e^(i theta_2))R1eiθ1R2eiθ2
= (R_1)/(R_2) e^(i (theta_1- theta_2))=R1R2ei(θ1−θ2)
R_1 = sqrt ( (2)^2 + (-1)^2) = sqrt 5R1=√(2)2+(−1)2=√5
R_2 = sqrt (4^2 + 1^2) = sqrt 17R2=√42+12=√17
tan theta_1 = - 1/2tanθ1=−12
tan theta_2 = 1/4tanθ2=14
From
tan(alpha-beta) = (tanalpha-tanbeta)/(1+tanalphatanbeta)tan(α−β)=tanα−tanβ1+tanαtanβ
tan (theta_1 - theta_2) =( (-1/2) - 1/4 ) / ( 1 + (-1/2)1/4 ) = - 6/7tan(θ1−θ2)=(−12)−141+(−12)14=−67
Which means that cos (theta_1 - theta_2) = 7 / sqrt 85cos(θ1−θ2)=7√85 and sin (theta_1 - theta_2) = -6 / sqrt 85sin(θ1−θ2)=−6√85
So
(R_1 e^(i theta_1))/(R_2 e^(i theta_2))R1eiθ1R2eiθ2
= (sqrt 5)/(sqrt 17) e^(i (arctan -6/7))=√5√17ei(arctan−67)
= (sqrt 5)/(sqrt 17) ( 7 / sqrt 85 - i 6 / sqrt 85 ) =√5√17(7√85−i6√85)
= 1/ (17) (7 - 6 i)=117(7−6i)
It's a lot simpler by finding the complex conjugate of the denominator d^prime and multiply the whole thing by the by (d^\prime) / (d^prime). As follows
(2 - i)/(4 + i) * (4 - i)/(4 - i)
= (8 - 2i - 4 i -1)/(16 - 4i + 4 i + 1)
= (7 - 6 i)/(17)
= 1/ (17) (7 - 6 i)
