How do you write an equation with solutions of 7 and -2

You can do this easily by working backwards. Quadratic equations commonly have two answers, so that's a good one to make.
Start by putting those answers into two factors, since you want a quadratic equation to equal 0, at least one of your factors has to be 0. Setting it up the way I have it below ensures one will be 0 with your given answers.

(x - 7)(x + 2) = 0   Multiply these together using the F.O.I.L. Method.

x² + 2x - 7x - 14 = 0   Simplify.

x² - 5x - 14 = 0