Within each set of parentheses, you should combine like terms—that is, add up all the terms that don't have $t$, that just have $t$, that have $t^2$, etc.
Once you've done that, each set of parentheses contains two terms being added (or subtracted). From the Distributive Property of Multiplication over Addition, $a(b+c)=ab+ac$ (the multiplication by $a$ is "distributed" to each of the terms $b$ and $c$ being added), $(a+b)(c+d)=$$(a+b)c+(a+b)d=$$ac+bc+ad+bd$ and $(a+b)^2=$$(a+b)(a+b)=$$a^2+ab+ab+b^2=$$a^2+2ab+b^2$. You can use this pattern to "expand" the rest of the squares in your equation.
Then, combine like terms on each side.
Finally, subtract the entirety of one side from both sides of the equation to get $=0$.
If the paper is essential reading and you're only stumbling over the symbolic manipulations like these, you might consider trying to use a Computer Algebra System (CAS) to help with some of the symbolic manipulation.