Final Answer:
The binomial 9x^2 + 49 cannot be factored using real numbers.
Step-by-step explanation:
The expression 9x^2 + 49 represents a sum of squares, specifically the square of 3x and the square of 7. It's in the form of a sum of squares which cannot be factored further using real numbers. When factoring quadratic expressions, the goal is often to express them as a product of two binomials. However, in this case, there are no two real numbers that can be multiplied together to yield 9x^2 + 49.
The expression is an example of a perfect square trinomial, which doesn't factor further into two linear binomials with real coefficients. Factoring typically involves breaking an expression down into simpler components, but in this scenario, the expression already consists of irreducible terms.
Upon applying the difference of squares rule (a^2 - b^2 = (a + b)(a - b)), it's evident that the given expression doesn't conform to this pattern. In the case of 9x^2 + 49, neither term is a perfect square multiple of the other, prohibiting its factorization using real numbers.
Therefore, the final answer remains that 9x^2 + 49 cannot be factored further in terms of real numbers, and it stands as an irreducible expression in the realm of real numbers.