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I am trying to find a polynomial in $\Bbb{Q}[x]$ which is irreducible over $\Bbb{Q}$ and has at least one linear factor over $\Bbb{R}$ and at least one irreducible quadratic factor over $\Bbb{R}$. Any help would be greatly appriciated.

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$$X^3-2{}{}{}{}{}{}{}{}{}{}{}$$

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$X^3-2 = (X-\alpha)(X^2+\alpha x+\alpha^2)$, where $\alpha =\sqrt[3]{2}$. –  lhf Mar 7 '13 at 17:07
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