Question 206019
{{{ x^2 - (3x^1) / 2 - 10 = 0 }}}  notation housekeeping
{{{ x^2 - (3/2)x - 10 = 0 }}} multiply both sides by 2 to get rid of the fraction
{{{ 2x^2 - 3x - 20 = 0 }}}

There are various ways of factoring a quadratic (if it factors, which this does, because {{{b^2-4*a*c = 9 + 160 = 169 }}} is a perfect square).  I like this way: 
{{{ 2x^2 - 3x - 20 = 0 }}} First multiply the coefficient of the {{{ x^2 }}} term, 2,  by the constant term, -20, to get -40.  Now you want to find two numbers that multiply to this number, -40, and that add to the coefficient of x, in this case -3. The two numbers are -8 and 5.
Now you put those two numbers into the blanks in {{{ 2x^2 + _x  + _x - 20 }}} (and it doesn't matter which number in which blank):
 
Case 1: {{{ 2x^2 + (-8)x  + (5)x - 20 }}}.  Now factor out whatever you can out of the first two terms and factor out whatever you can out of the last two terms:
{{{2x(x - 4) + 5(x - 4)}}}.  Notice both terms have a factor of x-4 which you can factor out:
{{{(2x + 5) (x-4)}}}.  
 
Case 2: {{{ 2x^2 + (5)x  + (-8)x - 20 }}}.  Now factor out whatever you can out of the first two terms and factor out whatever you can out of the last two terms:
{{{x(2x +5) + -4(2x + 5)}}}.  Again, both terms have a factor, 2x + 5 this time, which you can factor out:
{{{(x-4) (2x + 5)}}}. 
 
In both cases the factors are the same except in opposite order, which makes not a whit of difference.
It is a good idea to multiply out (FOIL) to make sure you have factored correctly
 
Now we use the zero product property which says that if a product of a bunch of numbers is 0 then one of the numbers must be 0, which is what allows us to set each of the factors to 0.
{{{(x-4) (2x + 5) = 0}}} --> {{{x-4 = 0}}} or {{{2x+5 = 0}}}, so {{{x = 4}}} or {{{x = -5/2}}}.  

Now it's a good idea to check your answers to make sure they work, by plugging them in (one at a time) for x in the ORIGINAL equation (I will do one, you can do the other):
{{{ (-5/2)^2 - (3(-5/2)^1) / 2 - 10 = 0 }}} ?
{{{ 25/4 + 15/4 - 10 = 0}}}?
{{{40/4 - 10 = 0}}}?
{{{0 = 0}}} Yes!