I assume you mean
[tex] \dfrac{5 + i}{1 + i} [/tex]
although what you wrote means something else.
To get rid of the imaginary part in the denominator, you must multiply this fraction by a fraction which is the complex conjugate of the denominator over itself.
The denominator of the original fraction is 1 + i.
Its complex conjugate is 1 - i.
Multiply the original fraction by
[tex] \dfrac{1 - i}{1 - i} [/tex]
Now we'll do it.
[tex] \dfrac{5 + i}{1 + i} \times \dfrac{1 - i}{1 - i} = [/tex]
[tex] = \dfrac{(5 + i)(1 - i)}{(1 + i)(1 - i)} [/tex]
[tex] = \dfrac{5 - 5i + i - i^2}{1 - i^2} [/tex]
[tex] = \dfrac{5 - 4i - (-1)}{1 - (-1)} [/tex]
[tex] = \dfrac{6 - 4i}{2} [/tex]
[tex] = 3 - 2i [/tex]
