Final answer:
To factorise the quadratic expression m^2 - 2mn - rn - 9r^2, we identify that it factorises to (m - 3r)(m + 3r) as these factors multiply to give the original expression.
Step-by-step explanation:
The question asks us to factorise the quadratic expression m^2 - 2mn - rn - 9r^2. To factorise this expression, we will look for two numbers that multiply to give the product of the coefficient of m^2 (which is 1) and the constant term -9r^2, and at the same time, they must add to give the coefficient of the middle term -2mn - rn.
By grouping terms and factoring by grouping, we identify that the two numbers we are looking for are -3r and +3r. This gives us the factors (m - 3r)(m + 3r).