Regular expression containing lambda
To generate an empty string you take: $(b+c)^*$ as $\epsilon$, and all the following elements which have a Kleene-star as $\epsilon$ (I will use $\epsilon$ instead of $\lambda$).

Then you have: $$\epsilon(a+\epsilon)\epsilon(a+\epsilon)\epsilon$$
Which is equivalent to: $$(a+\epsilon)(a+\epsilon)$$

Which is essentially picking two options a or $\epsilon$ twice back to back. If you take $\epsilon$ transition each time, which is actually a transition that does not consume an input, it will essentially accept an empty string as input.