the expression 3+i / 4+i is equivalent to
Multiply 3+i / 4+i by 1 by using the conjugate of 4+i over itself (4-i / 4-i)
(3+i)/(4+i) × (4-i)/(4-i)
(12-3i+4i-\(i^{2}\)) / (16-4i+4i-\(i^{2}\))
(12+i-\(i^{2}\)) / (16 - \(i^{2}\))
(12+i-(-1)) / (16-(-1))
(12+i+1) / (16+1)
13+i / 17
3+i / 4+i = \(\frac{13}{17} + \frac{1}{17}i\)
\(i=\sqrt{-1}\to i^2=-1\\========================\\\dfrac{3+i}{4+i}=\dfrac{3+i}{4+i}\cdot\dfrac{4-i}{4-i}=\dfrac{(3+i)(4-i)}{(4+i)(4-i)}=\dfrac{(3)(4)-(3)(i)+(i)(4)-(i)(i)}{4^2-i^2}\\\\=\dfrac{12-3i+4i-(-1)}{16-(-1)}=\dfrac{13+i}{17}=\boxed{\dfrac{13}{17}+\dfrac{1}{17}i}\)