<s>[INST] How to factor quadratic $ax^2+bx+c$? How do I shorten this? How do I have to think?
$$ x^2 + x - 2$$
The answer is 
 $$(x+2)(x-1)$$
I don't know how to get to the answer systematically. Could someone explain?
Does anyone have a link to a site that teaches basic stuff like this? My book does not explain anything and I have no teacher; this is self-studies.
Please help me out; thanks!
 [/INST] Still one more way: if you don't want to remember the quadratic formula, then just remember how it is proved. Let's do an example: Factor $3x^2 + 4x - 5$.
1) What gives $3x^2$ when squared? Answer: $\sqrt{3}x$.
2) What number $a$ will I have to add to this to get $4x$ as one of the terms in $(\sqrt{3}x + a)^2$? Answer: The coefficient of $x$ in $(\sqrt{3}x + a)^2$ is $2\sqrt{3}a$, so the answer is $\frac{2}{\sqrt{3}}$.
3) What is $\left(\frac{2}{\sqrt{3}}\right)^2$? It is $\frac{4}{3}$.
From these considerations we get
$$3x^2 + 4x - 5 = (\sqrt{3}x + \frac{2}{\sqrt{3}})^2 - \frac{4}{3} - 5 = (\sqrt{3}x + \frac{2}{\sqrt{3}})^2 - \frac{19}{3}.$$
Finally you can use the factorization $a^2 - b^2 = (a+b)(a-b)$ to conclude
$$3x^2 + 4x - 5 = (\sqrt{3}x + \frac{2+\sqrt{19}}{\sqrt{3}})(\sqrt{3}x + \frac{2-\sqrt{19}}{\sqrt{3}}).$$
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