Question

x²+5x+8; x²-5x+8; x²-5x-8; x²+5x-8; (x - 5x - 8)

Explanation:

Step1: Identify the problem type

The problem involves factoring quadratic expressions (e.g., \(x^2 + 5x + 8\), \(x^2 - 5x + 8\), etc.). This falls under Algebra, a subfield of Mathematics.

Step2: Analyze the quadratic form

For a quadratic \(ax^2 + bx + c\) (here \(a = 1\) for all shown), we check if it factors. Let's take \(x^2 + 5x + 8\): find two numbers that multiply to \(8\) and add to \(5\). Factors of \(8\): \(1\&8\) (sum \(9\)), \(2\&4\) (sum \(6\))—no integer factors. For \(x^2 - 5x + 8\): numbers multiplying to \(8\), adding to \(-5\)—negative factors: \(-1\&-8\) (sum \(-9\)), \(-2\&-4\) (sum \(-6\))—no integer factors. \(x^2 - 5x - 8\): multiply to \(-8\), add to \(-5\)—factors of \(-8\): \(1\&-8\) (sum \(-7\)), \(2\&-4\) (sum \(-2\))—no. \(x^2 + 5x - 8\): multiply to \(-8\), add to \(5\)—factors: \(8\&-1\) (sum \(7\)), \(4\&-2\) (sum \(2\))—no. So these quadratics don't factor over integers. If the question was to factor, the answer is they are irreducible over integers.

Answer:

The quadratic expressions \(x^2 + 5x + 8\), \(x^2 - 5x + 8\), \(x^2 - 5x - 8\), \(x^2 + 5x - 8\) are irreducible (cannot be factored) over the set of integers.