First we need to simplify the numerator of the fraction
(2 - 4i)(3 + 5i)
= 6 +10i - 12i - 20i²
= 6 - 2i + 20
= 26 - 2i
So now the fraction becomes:
[tex] \frac{26-2i}{3+i} [/tex]
In order to convert the fraction to standard form, we need to multiply and divide the fraction by the conjugate of the denominator as shown below:
[tex] \frac{26-2i}{3+i}* \frac{3-i}{3-i} \\ \\
= \frac{78-26i-6i+2i^{2} }{9-i^{2} } \\ \\
= \frac{78-32i-2}{9+1} \\ \\
= \frac{76-32i}{10} \\ \\
= \frac{76}{10}- \frac{32i}{10} \\ \\
= \frac{38}{5}- \frac{16}{5}i [/tex]
