\[{x^3} - 2x + 4 = \sqrt {{x^2} + 4x + 4} \]

\[ \Leftrightarrow {x^3} - 2x + 4 = \sqrt {{{(x + 2)}^2}} \]

\[ \Leftrightarrow {x^3} - 2x + 4 = \left| {x + 2} \right|\]

Vì x > 0 nên x + 2 > 0

Û x³ – 2x + 4 = x + 2

Û x³ – 2x + 4 – x – 2 = 0

Û x³ + 8 – 2x – 4 – x – 2 = 0

Û (x + 2)(x² – 2x + 4) – 2(x + 2) – (x + 2) = 0

Û (x + 2)(x² – 2x + 4 – 2 – 1) = 0

Û (x + 2)(x² – 2x + 1) = 0

Û (x+2)(x – 1)² = 0

\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}{x + 2 = 0}\\{{{(x - 1)}^2} = 0}\end{array}} \right. \Rightarrow \left[ {\begin{array}{*{20}{c}}{x + 2 = 0}\\{x - 1 = 0}\end{array}} \right. \Rightarrow \left[ {\begin{array}{*{20}{c}}{x = - 2\,\,\,(L)}\\{x = 1\,\,\,(tm)}\end{array}} \right.\]

Vậy x = 1 là nghiệm của phương trình đã cho.