Write the function as #f(x)=(x+1)sqrt{x^2+2x+5}# and then use the Product Rule and the Chain Rule to get the first derivative:
#f'(x)=sqrt{x^2+2x+5}+(x+1) * 1/2 * (x^2+2x+5)^{-1/2} * (2x+2)#
#=(x^2+2x+5+x^2+2x+1)/sqrt{x^2+2x+5}=(2x^2+4x+6)/sqrt{x^2+2x+5}#
Now use the Quotient Rule and Chain Rule to find the second derivative:
#f''(x)#
#=(sqrt{x^2+2x+5} * (4x+4)-(2x^2+4x+6) * 1/2 * (x^2+2x+5)^{-1/2} * (2x+2))/(x^2+2x+5)#
#=((x^2+2x+5)(4x+4)-(2x^2+4x+6)(x+1))/((x^2+2x+5)^{3/2})#
#=(4x^3+8x^2+20x+4x^2+8x+20-2x^3-4x^2-6x-2x^2-4x-6)/((x^2+2x+5)^{3/2})#
So
#f''(x)=(2x^3+6x^2+18x+14)/((x^2+2x+5)^{3/2})#
Since #x^2+2x+5>0# for all real numbers #x#, the sign (positive or negative) of #f''(x)# is determined by the sign of #2x^3+6x^2+18x+14#, which is determined by the sign of #x^3+3x^2+9x+7#.
The number #x=-1# is a real root of #x^3+3x^2+9x+7# since #-1+3-9+7=0#. If you use long-division of polynomials (or synthetic division), you'll find that #x^3+3x^2+9x+7=(x+1)(x^2+2x+7)#. The quadratic #x^2+2x+7# is always positive for all real #x#.
All of this implies that the sign of #f''(x)# is determined by the sign of #x+1#. Since #x+1>0# when #x > -1# and #x+1<0# when #x < -1#, the same is true for #f''(x)#.
This means the graph of #f(x)# is concave up for #x > -1# (on the interval #(-1,infty)#) and concave down for #x < -1# (on the interval #(-infty,-1)#).
