Simplify fully (3x^2-8x-3)/(2x^2-6x)

While this can appear intimidating at first, the key to simplifying this expression is through factorising and identifying common factors. Let's start off by factorising the denominator as this is simpler. How can we simplify (2x^2-6x)? 2x is a factor of each of the components of the expression, therefore we can take the 2x out, leaving us with 2x(x-3). There we go, (x-3) must be one of the factorised components of the quadratic equation in the numerator because we can assume that they will cancel out. So, when factorising (3x^2-8x-3), it will look like (x-3)(?x+?). The coefficient of the x^2 term is 3, leaving us with (x-3)(3x-?). If we expand the brackets, we get (3x^2-?x-9x-3). The coefficient of x in the numerator is -8, therefore (?-9)x must be equal to -8x. We know that 1-9 = -8. Therefore, leaving us with (x-3)(3x+1) - we have now factorised the numerator! Because the (x-3) is a common factor in the numerator and denominator - this can now be cancelled out, giving (3x+1)/(2x) as the final answer.