For this case we have the following functions:
[tex]f (x) = x ^ 2 + 6x + 5\\g (x) = \frac {1} {x + 1}[/tex]
We must find [tex](f_ {0} g) (x).[/tex] By definition of composition of functions we have to:
[tex](f_ {0} g) (x) = f (g (x))[/tex]
So, we have:
[tex]f (g (x)) = (\frac {1} {x + 1}) ^ 2 + 6 (\frac {1} {x + 1}) + 5\\f (g (x)) = \frac {1} {(x + 1) ^ 2} + \frac {6} {x + 1} +5\\f (g (x)) = \frac {1 + 6 (x + 1) +5 (x + 1) ^ 2} {(x + 1) ^ 2}\\f (g (x)) = \frac {1 + 6 (x + 1) +5 (x + 1) ^ 2} {(x + 1) ^ 2}\\f (g (x)) = \frac {1 + 6x + 6 + 5 (x ^ 2 + 2x + 1)} {(x + 1) ^ 2}\\f (g (x)) = \frac {1 + 6x + 6 + 5x ^ 2 + 10x + 5} {(x + 1) ^ 2}\\f (g (x)) = \frac {5x ^ 2 + 16x + 12} {(x + 1) ^ 2}\\f (g (x)) = \frac {(5x + 6) (x + 2)} {(x + 1) ^ 2}[/tex]
Answer:
[tex]f (g (x)) = \frac {(5x + 6) (x + 2)} {(x + 1) ^ 2}[/tex]
