Factor completely, or state that the polynomial is prime: x² + 25.

Step-by-step solution

1. 1

   Check for a common factor. The terms are $x^2$ and $25$; they share no variable and no numerical factor larger than $1$, so nothing can be pulled out.

2. 2

   Test the difference of squares pattern. That pattern needs a subtraction, $a^2 - b^2$. Here the terms are added, so $x^2 + 25$ is a sum of squares and the pattern does not apply.

3. 3

   Look for real roots. Setting $x^2 + 25 = 0$ gives $x^2 = -25$, which no real number satisfies. A real quadratic with no real roots cannot be written as a product of two real linear factors.

4. 4

   Confirm with the discriminant. For $x^2 + 0x + 25$ the discriminant is $0^2 - 4(1)(25) = -100 < 0$, the algebraic statement of the same fact.

5. 5

   Conclude. Over the real numbers $x^2 + 25$ is prime. (Only if complex factors are allowed does it become $(x - 5i)(x + 5i)$.)