Solve the inequality x^2 > 3(x + 6)

In order to solve this inequality, it is helpful to have all the terms on the left-hand side of the equation. To do this, we can subtract 3(x + 6) from both sides of the equation. This gives us:

x^2 - 3(x+6) > 0 

Now, take notice of the brackets - we want to get rid of those in order to solve this. To do this, we 'expand' the brackets by multiplying the terms inside the brackets by the terms that the brackets are multiplied by. In this case, we have x+6 inside the brackets and 3 outside the brackets. -3 * (x + 6) equals -3x - 18. Take special care to note that the minus sign when it appears.

So we now have 

x^2 - 3x - 18 

We can now factorise this, which lets us see which values of x result in 0, which helps us solve this inequality. 

For this case, we can factorise it as follows

(x + a)(x + b) > 0

Where ab = -18 and a + b = -3

If we look at the factors of -18, we can see that 3 and -6 add up to -3, satisfying both of these equations. So:

(x - 6)(x+3) > 0

Plotting these on a graph, we have a convex ('u' shaped) curve with intercepts at x = 6 and x = -3, and we can see that this is greater than 0 where x > 6 and x < -3, which is our answer.