"both the ideas" is grammatically correct and can be used in written English.
It is used to refer to two ideas that are being compared or contrasted. Example: Both the ideas presented in the meeting are worth considering, but I think the second one is more feasible.
They added, "The Times has both the ideas and the money to pursue innovation".
The Times has both the ideas and the money to pursue innovation".
Conservationists are heatedly debating that shift, since both the ideas behind it – and its possible implications – remain controversial.
Both the ideas and the terminology of acid base chemistry have permeated daily life, and the term salt is especially common.
The double encryption method detailed in previous section incorporates both the ideas of symmetric and asymmetric cryptosystem efficiently and minimizes most of the shortcomings associated with both approaches.
If one is an imagist it becomes impossible to imagine what idea could include both the ideas of a right and equilateral triangle.
David Cameron finds both the idea of Spitzenkandidaten, and the politics of the two frontrunners, unpalatably federalist.
True, last season Contact! had the advantage of novelty: both the idea and the works performed were freshly minted.
Both the idea that feminism needs to be appealing to men and that men should be central to its progress are anti-feminist ideas in themselves.
He thinks the city may find itself reconsidering both the idea of allowing housing in North Bayshore and the amount of space devoted to commercial use.
Republicans object to both the idea of a new government plan and the use of expedited procedures to push it through the Senate on a simple majority vote.
