The expression in the image suggests that we need to find a monomial that, when squared, results in the given identity:

$( \ldots + 2b)^2 = a^2 + 4ab + 4b^2$

To match the form on the right-hand side, let's expand the square of a binomial:

$(x + 2b)^2 = x^2 + 2 \cdot x \cdot 2b + (2b)^2$

This simplifies to:

$x^2 + 4bx + 4b^2$

For the expression on the right to match $a^2 + 4ab + 4b^2$, we can deduce that $x = a$. Therefore, the correct monomial to insert is $a$.

Thus, the identity will be:

$(a + 2b)^2 = a^2 + 4ab + 4b^2$

Let me know if you'd like further details or have any questions!

Here are some related questions to consider:

1. What is the general expansion rule for squaring binomials?
2. How can the distributive property be used to expand polynomials?
3. What is the importance of recognizing binomial expansions in algebra?
4. How do we simplify expressions that involve squares and like terms?
5. Can you apply the same method to expand a trinomial or more complex polynomial?

Tip: Always look for patterns when expanding binomials, as it helps in matching the desired form.