<s>[INST] How to factor a fourth degree polynomial I'm working on a math problem but I am having a hard time figuring out the method used by my textbook to make this factorization:
$$x^4 + 10x^3 + 39x^2 + 70x + 50 = (x^2 + 4x + 5)(x^2 + 6x + 10)$$
I've tried to see if this equation can be factored by grouping or by long division to no avail. Any help would be greatly appreciated.
 [/INST] Let
$$f (x) = x^4 + 10 x^3 + 39 x^2 + 70 x + 50$$
Converting to a depressed quartic see here, we see that the $x$ term drops out as well
$$f\left(x-\frac{5}{2}\right)=x^4+\frac{3 x^2}{2}+\frac{25}{16}=\left(x^2+\frac{5}{4}\right)^2-x^2=\left(x^2+\frac{5}{4}+x\right)\left(x^2+\frac{5}{4}-x\right)$$
The factorization of $f$ is obtained by replacing $x$ with $x+\frac{5}{2}$
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