\(\dfrac{a+4\sqrt{a}+4}{\sqrt{a}+2}+\dfrac{4-a}{2-\sqrt{a}}\)
\(=\dfrac{\left(\sqrt{a}+2\right)^2}{\sqrt{a}+2}+\dfrac{\left(2-\sqrt{a}\right)\left(2+\sqrt{a}\right)}{2-\sqrt{a}}\)
\(=\sqrt{a}+2+2+\sqrt{a}=2\sqrt{a}+4\)
\(\dfrac{x+1-2\sqrt{x}}{\sqrt{x}-1}+\dfrac{x+\sqrt{x}}{\sqrt{x}+1}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}-1}+\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\)
\(=\sqrt{x}-1+\sqrt{x}=2\sqrt{x}-1\)
`(a + 4 sqrt{a} + 4)/(sqrt{a} + 2) + (4-a)/(2-sqrt{a})`
`= ((sqrt{a} + 2)^2)/(sqrt{a} + 2) + ((2-sqrt{a})(2+sqrt{a}))/(2-sqrt{a})`
`= sqrt{a} + 2 + 2+sqrt{a}`
`= 4+2sqrt{a}`
`(x + 1 - 2sqrt{x})/(sqrt{x} - 1) + (x+sqrt{x})/(sqrt{x} + 1)`
`= ((sqrt{x} - 1)^2)/(sqrt{x} - 1) + (sqrt{x}(sqrt{x} + 1))/(sqrt{x} + 1)`
`= sqrt{x} - 1 + sqrt{x}`
`= 2sqrt{x} - 1`
