Prove the following by using the principle of mathematical inductio...

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Prove the following by using the principle of mathematical induction for all \( n \in \mathbf{N} \) :
\[
1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3} .
\]
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