y^2=4^2+x^2. Take square roots of each side: y=√(16+x^2), which is the same as given.
You have assumed that the square of x+4 is x^2+16, or that the square root of x^2+16 is x+4. The square of x+4=x^2+8x+16. (x+4)^2=(x+4)(x+4)=x^2+4x+4x+16=x^2+8x+16. So, in your reasoning, 8x would be unaccounted for.
You can prove this for yourself by example: Put x=3: y^2=4^2+3^2=16+9=25 so y=5; but x+4 would be 7, according to your reasoning.