An inventory analyst is auditing a calculation used for determining storage requirements, which includes the expression $n^2 + 64$. To simplify the data entry, the analyst is asked to factor this expression into a product of binomials. After confirming there is no greatest common factor, what should the analyst conclude about $n^2 + 64$?

A project manager estimating material costs for two square flooring sections models the total area with the expression $x^2 + 25$. After confirming there is no greatest common factor, she notes in her spreadsheet that because the expression is a sum of squares, it cannot be factored into a product of binomials and is therefore considered ____.

A landscape architect is calculating the combined area of two square fountain basins, which is represented by the expression ${}49 + y^2$. This expression is considered prime because, after confirming there is no greatest common factor, a sum of squares cannot be factored into a product of binomials.

Factoring Constraints in Risk Modeling

An inventory manager is auditing the algebraic models used to determine the storage capacity of square-based shipping bins. To organize the database, the manager needs to distinguish between expressions that can be factored and those that are 'prime' (cannot be factored). Match each binomial expression with its correct classification regarding its factorability.