One of the factors of

$\left(25 x^{2}-1\right)+(1+5 x)^{2}$

$=(5 x-1)(5 x+1)+(1+5 x)^{2}$

$=(5 x+1)[(5 x-1)+(1+5 x)]$

$=(5 x+1)(10 x)$

So, the factors of $\left(25 x^{2}-1\right)+(1+5 x)^{2}$ are $(5 x+1)$ and $10 x$

Hence, the correct answer is option (d).