To factor the quadratic expression \(x^2 + x + 12\), we need to find two numbers that multiply to \(12\) (the constant term) and add up to \(1\) (the coefficient of the \(x\) term).
After checking the possible pairs of factors of \(12\):
- \(1 \times 12\)
- \(2 \times 6\)
- \(3 \times 4\)
None of these pairs add up to \(1\).
Since there are no two numbers that meet these criteria, the expression \(x^2 + x + 12\) cannot be factored using real numbers.
Thus, it is irreducible.
In conclusion, \(x^2 + x + 12\) does not factor neatly, and it is generally left in its original form unless using complex numbers.
If you want the expression in its non-factorable form, you can say:
Answer: \( (x^2 + x + 12) \) (cannot be factored over the reals)
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