To determine a common factor between the two algebraic expressions, we need to factor each expression separately.
First, let’s factor the expression x² + 6. This expression cannot be factored using real numbers, as it does not have any rational roots. Hence, it remains in its simplest form.
Now, let’s consider the second expression, x² + 5x + 6. We can factor this expression by finding two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the linear term). The numbers 2 and 3 fit this requirement.
Thus, we can factor x² + 5x + 6 as follows:
- x² + 5x + 6 = (x + 2)(x + 3)
Now, observing the two expressions:
- x² + 6 (unfactored)
- (x + 2)(x + 3) for x² + 5x + 6
Since x² + 6 does not share any common factors with x² + 5x + 6, the answer is:
There is no common factor between the two expressions in terms of linear factors. The first expression is irreducible in real numbers.