How do you factorise?

Factorising is taking an expression of some sort and simplifying it down to its component parts so that they are presented in the simplest way possible. For example, take the expression (15pq + 10pqr). First we look for common factors between the two terms within the bracket. Both numbers are divisible by 5 so we divide and take 5 outside the brackets to get 5(3pq + 2pqr). Both parts of the expression now also have the pq element. We do the same as before, dividing both parts by pq leaving 5pq(3+2r). The terms within the bracket now have no common factors, so we cannot simplify any further and the expression has been factorised. This can get a bit more tricky when quadratics are involved, but we still look for common factors. Take the example 3x^2 + 15x + 18. Initially, we notice that all the terms are divisible by 3. We get: 3(x^2 + 5x + 6). The term within the bracket is now a simpler quadratic to factorise. We are looking to get this into two brackets that have been multiplied together, usually in the form (x ± ?)(x ± ?). My trick is to find factors of the last number (the constant) which add up to the coefficient of the middle term. In this case, 2 and 3 are both factors of 6 and sum to 5. We would get 3(x+3)(x+2). A quick check and we see that this is also the fully factorised form.