1. The quadratic \(x^2+4x+3\) factorises as \((x+1)(x+3)\). In both the original quadratic and the factorised form, all of the coefficients are integers.

    The quadratic \(x^2-4x+3=(x-1)(x-3)\) similarly factorises with all of the coefficients being integers.

    How many quadratics of the following forms factorise with integer coefficients? Here, \(b\) is allowed to be any integer (positive, negative or zero). For example, in (a), \(b\) could be \(-7\), since \((x-2)(x-5)=x^2-7x+10\).

    1. \(x^2+bx+10\)

    What happens when you multiply out a pair of brackets?

    Alternatively, if we knew the value of \(b\), how would we go about attempting to factorise the quadratic?

    How might this help us to work out which values of \(b\) allow us to factorise the quadratic?

If you need a reminder of how to factorise quadratics where the \(x^2\) term has a coefficient other than \(1\), you may wish to look at the Quadratic grids resource.