To factor the expression x² + 6, we need to look for two numbers that multiply to give 6 and add to give 0, the coefficient of the x term. However, we quickly notice that there are no such integer pairs.
This expression can, however, be seen as a sum of squares, since it can be rewritten as:
x² + (√6)²
Factorizing a sum of squares directly is not possible using real numbers; we only get the difference of squares using the following identity:
(a² – b²) = (a + b)(a – b).
Therefore, when it comes to real numbers, we can conclude that x² + 6 does not factor nicely like x² – 6 does into real factors. If we venture into complex numbers, we can express it as:
x² + 6 = (x – i√6)(x + i√6)
In summary, x² + 6 is not factorable over the rationals or reals, but we can factor it over the complex numbers as shown above.