Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as
$$320 x ^ { 3 } + k x$$
where the value of the constant \(k\) is to be stated.
Verify that \(x = 2\) is a root of the equation
$$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$
and find the other possible values of \(x\).