We start with the polynomial expression:
\[
25 v^{2} + 40 v + 16
\]
Step 2: Factor the Polynomial
To factor the polynomial, we look for two identical binomials. The expression can be rewritten as:
\[
(5v + 4)^{2}
\]
Step 3: Verify the Factorization
To ensure the factorization is correct, we can expand \((5v + 4)^{2}\):
\[
(5v + 4)(5v + 4) = 25v^{2} + 20v + 20v + 16 = 25v^{2} + 40v + 16
\]
This confirms that the factorization is accurate.