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Answer
\( x^{2} + x + 12 \) cannot be factored into real numbers because it has no real roots.
Solution
The expression \( x^{2} + x + 12 \) does not factor nicely into real numbers because its discriminant is negative.
To confirm this, we can calculate the discriminant \( D \) using the formula:
\[
D = b^2 - 4ac
\]
where \( a = 1 \), \( b = 1 \), and \( c = 12 \).
Calculating the discriminant:
\[
D = 1^2 - 4 \cdot 1 \cdot 12 = 1 - 48 = -47
\]
Since the discriminant is negative, the quadratic does not have real roots and cannot be factored over the real numbers.
Thus, \( x^{2} + x + 12 \) is already in its simplest form and cannot be factored further. If you need the complex factorization, please let me know!
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Simplify this solution Beyond the Answer
To factor the expression \( x^2 + 1x + 12 \), we look for two numbers that multiply to 12 (the constant term) and add to 1 (the coefficient of \( x \)). However, there are no integer pairs that satisfy both conditions. Thus, this quadratic does not factor nicely over the integers. Instead, you could use the quadratic formula to find its roots, or you could leave it expressed as \( (x^2 + 1x + 12) \) since it doesn't simplify further.
