This isn't a "shorter way" per se, but is an alternative if you're really that lazy to expand...
Let

. Since

is a root of the polynomial then
Since we get same result regardless of whether we define y in terms of

then this new equation has the roots

.
Hence
(2\beta+1)(2\gamma+1))
is just the product of the roots of that new equation.
By inspection of the expansion we can see that the constant term is -1- 4-16+56, or equivalently 35. The product of the roots is therefore -35.