<s>[INST] Factor $6x^2​ −7x−5=0$ I'm trying to factor
$$6x^2​ −7x−5=0$$
but I have no clue about how to do it. I would be able to factor this:
$$x^2-14x+40=0$$
$$a+b=-14$$
$$ab=40$$ 
But $6x^2​ −7x−5=0$ looks like it's not following the rules because of the coefficient of $x$. Any hints?
 [/INST] Here's an alternative (but you should just learn the quadratic equation)
$6x^2 - 7x - 5= 6(x + a)(x + b)$ 
So $ab = -5/6$ and $a + b = -7/6$.
$a = \pm 1, \pm 5, \pm 1/2, \pm 5/2, \pm 1/3, \pm 5/3, \pm 1/6, \pm 5/6$
$b = \mp 5/6, \mp 1/6, \mp 5/3, \mp 1/3, \mp 5/2, \mp 1/2, \mp 5, \mp 1$.
So $a + b = \pm 2/3, \pm 29/6, \mp 7/6, \pm 13/6, etc....$
So $a = 1/2$ and $b= -5/3$ works.
$6(x + 1/2)(x - 5/3) = (2x + 1)(3x - 5) = 6x^2 - 7x -5$.
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