All proofs by mathematical induction follow the same steps. Algebra skills are required.
1. Test if proposition is true for n=1. 1 times 3 does equal 1(4+6-1)/3.
2. Assume true for n=k.
3. Show that it follows, algebraically, that it must then be true for n = k+1, based on writing the statement to be proved, with n=k on each side, AND THEN ADDING the next term to both sides.
That means add (2k+1)(2k+3) to each side. Then YOU must factor the right side and show that this equation's two sides correspond to "proposition k+1" and thus, proposition k (assumed) implies the truth of proposition k+1. (Then this implies,by repeated applications, truth in general.)
So get going on the necessary algebra!