For the real numbers $a$ and $b$ satisfying $$a ^ { 3 } = \frac { 1 } { \sqrt { 5 } - 2 } , \quad b ^ { 3 } = 2 - \sqrt { 5 }$$ we are to find the value of $a + b$. When we set $x = a + b$, we have $$x ^ { 3 } = ( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + \mathbf { A } a b ( a + b ) .$$ Since $a b = \mathbf { B C }$, we know that this $x$ satisfies $$x ^ { 3 } + \mathbf { D } x - \mathbf { E } = 0 .$$ The left side of this equation can be factorized as follows: $$\begin{aligned}
x ^ { 3 } + \mathbf { D } x - \mathbf { E } & = \left( x ^ { 3 } - \mathbf { F } \right) + \mathbf { D } \left( x - \frac { \mathbf { F } } { \mathbf { F } } \right) \\
& = ( x - \mathbf { F } ) \left( x ^ { 2 } + x + \mathbf { G } \right) .
\end{aligned}$$ Since $$x ^ { 2 } + x + \mathbf { G } = \left( x + \frac { \mathbf { H } } { \mathbf { I } } \right) ^ { 2 } + \frac { \mathbf { J K } } { \mathbf { L } } > 0 ,$$ we obtain $x = a + b = \mathbf { M }$.
For the real numbers $a$ and $b$ satisfying
$$a ^ { 3 } = \frac { 1 } { \sqrt { 5 } - 2 } , \quad b ^ { 3 } = 2 - \sqrt { 5 }$$
we are to find the value of $a + b$.
When we set $x = a + b$, we have
$$x ^ { 3 } = ( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + \mathbf { A } a b ( a + b ) .$$
Since $a b = \mathbf { B C }$, we know that this $x$ satisfies
$$x ^ { 3 } + \mathbf { D } x - \mathbf { E } = 0 .$$
The left side of this equation can be factorized as follows:
$$\begin{aligned}
x ^ { 3 } + \mathbf { D } x - \mathbf { E } & = \left( x ^ { 3 } - \mathbf { F } \right) + \mathbf { D } \left( x - \frac { \mathbf { F } } { \mathbf { F } } \right) \\
& = ( x - \mathbf { F } ) \left( x ^ { 2 } + x + \mathbf { G } \right) .
\end{aligned}$$
Since
$$x ^ { 2 } + x + \mathbf { G } = \left( x + \frac { \mathbf { H } } { \mathbf { I } } \right) ^ { 2 } + \frac { \mathbf { J K } } { \mathbf { L } } > 0 ,$$
we obtain $x = a + b = \mathbf { M }$.