simplify fully: (3x^2 - 8x -3)/(2x^2 -6x)

First of all, to simplify this fraction, we need to factorise the top and bottom equations. We shall start with the top equation. Now looking at the equation: 3x² - 8x -3, we know that it's a quadratic with 3 different terms, so when we factorise this, it will look something like: ( _x +/- _ )( _x +/- _ ). We know that the two 'x' terms in the brackets need to multiply to make 3x², so the coefficients of each of the x terms in brackets will be 3 and 1 because 3x1=3. So we now know that the brackets will look more like: (3x +/- _ )(x +/- _ ). We know that the two terms without an 'x' must multiply to make -3, so these coefficients must either be: ( -3 and 1 ) or ( 3 and -1 ). Now we need to deduce how to obtain the -8x from the coefficients that we have, and this will depend on which set of coefficients for the none-x term that we pick, and where we place them in the brackets. From a bit of trial and error we will find that the brackets should look like this: (3x+1)( x-3 ) because when we expand this the 'x' term will be made from (3x) x (-3) +(x)(1) which gives -9x +1x which is -8x.
Now when we factorise the 2x² - 6x we simply get: 2x( x-3 ). So the fractions will look like: (3x+1)( x-3 )/2x( x-3 ). We can cancel out the ( x-3 ) terms on the top and the bottom of the fraction which will leave us with: (3x+1)/2x, which is the simplest version of this fraction.