Simplify fully the fraction (a^2 + a − 6)/ (ab+3b)

so, let's split the fraction up, into the top half and the bottom half, and deal with each part at a time. The top part of the fraction (numerator) is a^2 +a -6 , and you can see that it is a quadratic equation, as it has a squared a. The easiest way to simplify this would be to use factorisation. As the numerator is in the same form as ax^2 + bx+c, we need to find two numbers that would add together to make b (1) and multiply to make c (-6). These numbers are 3 and -2, therefore, the equation can be rewritten as (a+3)(a-2) and this is the simplest form that the numerator can take. Let's move onto the denominator, which is ab+3b This can be simplified by taking the common b out, making b(a+3) and this is the simplest form that the denominator can take. If we put the simplified top and bottom together again, we get (a+3)(a-2)/ b(a+3) As you can see, there is (a+3) common to both the top and the bottom, which we can cancel. This then simplifies this further to (a-2)/b , which is the simplest this fraction can get.