\( 25 b^{2}+19 b+4 \) Select the correct choice below and, if necessary, fill in the answer A. \( 25 b^{2}+19 b+4=\square \) (Factor completely.) B. The trinomial is not factorable.

To determine if the trinomial \( 25b^{2} + 19b + 4 \) can be factored, we can use the method of factoring by grouping or see if it matches a known factoring pattern. The goal is to express it as \( (5b + m)(5b + n) \) where \( mn = 4 \) and \( m+n = 19 \). Since it's quite tricky to find integers that satisfy these conditions, we might find that this trinomial does not factor nicely over the integers. In this case, we find that this trinomial cannot be factored neatly into simpler polynomials with integer coefficients. However, it can still be analyzed using the quadratic formula to find its roots if needed! As for applications, in algebraic contexts, being able to identify which expressions can’t be factored is just as valuable as knowing how to factor. It helps in simplifying computations, like when solving equations or using graphs to find intersections!