\[\begin{array}{l} {{\text{2}}^{10}} + {5^{12}} = {\left( {{2^5}} \right)^2} + {\left( {{5^6}} \right)^2} = {\left( {{2^5} + {5^6}} \right)^2} - 2 \cdot {2^5} \cdot {5^6} = \\ {\left( {{2^5} + {5^6}} \right)^2} - {2^6} \cdot {5^6} = {\left( {{2^5} + {5^6}} \right)^2} - {10^6} = \\ {\left( {{2^5} + {5^6}} \right)^2} - {\left( {{{10}^3}} \right)^2} = \left( {{2^5} + {5^6} - {{10}^3}} \right) \cdot \left( {{2^5} + {5^6} + {{10}^3}} \right) \end{array}\]