I'm learning about factoring and the lecturer show this example:
$$-3x^2+12x-18$$
For start he factor this polynomial like:
$$3(-x^2+4x-6)$$
So far so good but now he said:
In some circumstances it's more valuable to us to take out negative factor than positive factor
And now:
$$-3(x^2-4x+6)$$
I really don't know why should to prefer negative factor than positive factor, is there a special reason?
Yes, but it is a matter of getting the remaining polynomial to have a leading factor of $+1$. it is just a bit easier to factor a polynomial with leading coefficient $+1$. For example, a quadratic like $$ x^2 + bx + c$$ factors into $$(x+r)(x+s)$$ with $$rs = c \\ r+s = b$$ So if you have a hunch that the polynomial given will factor "nicely" you need only look for pairs of factors of the constant term.
For example, to factor $x^2+5x+6$ you only have to try $$ (x+1)(x+6) \\ (x+2)(x+3) \\ (x-1)(x-6) \\ (x-2)(x-3) $$ one of these must work if the polynomial has rational factors. In the case I showed, the second wone works because $2+3=5$.