To factor the quadratic expression [tex]\(x^2 - 4x + 5\)[/tex], let’s go through the standard factoring process step by step.
1. Identify the coefficients: In the quadratic [tex]\(x^2 - 4x + 5\)[/tex], the coefficient of [tex]\(x^2\)[/tex] is [tex]\(1\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-4\)[/tex], and the constant term is [tex]\(5\)[/tex].
2. Check for factors involving integers: Generally, factoring involves finding two binomials [tex]\((x - p)(x - q)\)[/tex] whose product is [tex]\(x^2 - 4x + 5\)[/tex]. Here, [tex]\(p\)[/tex] and [tex]\(q\)[/tex] should satisfy two conditions:
- Their product [tex]\(p \cdot q\)[/tex] should equal the constant term [tex]\(5\)[/tex].
- Their sum [tex]\(p + q\)[/tex] should equal the coefficient of [tex]\(x\)[/tex] term, which is [tex]\(-4\)[/tex].
However, the product [tex]\(p \cdot q = 5\)[/tex] and sum [tex]\(p + q = -4\)[/tex] do not match any pair of integers because 5 is a positive integer and there are no integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex] that satisfy these conditions at the same time.
3. Conclude on the factorability: Since it is not possible to factorize the quadratic equation [tex]\(x^2 - 4x + 5\)[/tex] over the real numbers using integers or simple rational numbers, it means the expression is prime.
Thus, the polynomial [tex]\(x^2 - 4x + 5\)[/tex] does not factor over the real numbers and remains in its original form.
Therefore, the correct selection is:
- Prime.
So, the detailed explanation is:
- The quadratic [tex]\(x^2 - 4x + 5\)[/tex] does not factor over the real numbers, so it is prime.
