\[\begin{array}{l}
\frac{{{a^3} + {b^3}}}{{{a^3} + {c^3}}} = \frac{{a + b}}{{a + c}}\\
\Leftrightarrow \frac{{\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}}{{\left( {a + c} \right)\left( {{a^2} - ac + {c^2}} \right)}} = \frac{{a + b}}{{a + c}}\\
\Leftrightarrow \frac{{{a^2} - ab + {b^2}}}{{{a^2} - ac + {c^2}}} = 1\\
\Leftrightarrow {a^2} - ab + {b^2} = {a^2} - ac + {c^2}\\
\Leftrightarrow {b^2} - ab = {c^2} - ac\\
\Leftrightarrow {b^2} - {c^2} + ac - ab = 0\\
\Leftrightarrow \left( {b - c} \right)\left( {b + c} \right) - a\left( {b - c} \right) = 0\\
\Leftrightarrow \left( {b - c} \right)\left( {b + c - a} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
b = c\\
a = b + c
\end{array} \right.
\end{array}\]
