Find the values of x that satisfy the quadratic equation: x^2 + 14x + 40 = 0

We use factorisation of double brackets to re-express the equation in a more useful form. For this factorisation, we require two numbers that add to make 14 and also multiply to make 40. We notice that the numbers 10 and 4 satisfy this condition.Hence we can factorise: so x^2 + 14x + 40 = (x+10)(x+4) and since the quadratic equation was set equal to 0 then: (x+10)(x+4) = 0.
For this equation to hold, either the first bracket must be equal to 0 or the second. In other words, either: (x+10) = 0 or (x+4) = 0 which means either: x = -10 or x= -4. (These values satisfy our original quadratic equation and we can simply check that this is correct by substituting x = -10 and x= -4 back into the quadratic equation and we will find that the expression indeed equals 0 in both cases).