sqrt{x^2 } +x^2/(x^2+1)

So we have

√(x²) + x² / [x² +1]    and we just want to simplifiy this

First of all,we need to be careful with the first term.....many people simplify this as just "x", but that's not correct. To see why, what if x = -3 ??? If we wrote "x," we would be claiming that we could get a negative answer from a positive square root, which is impossible. The correct answer is that √(x²) = lxl  ... (this is the definition of absolute value anyway....)

So we have

lxl +  x² / [x² +1]    and getting a common denominator, we have

[ lxl* (x² +1) +  x²  ] / [x² +1]