Binomial Theorem - Binomial Expansion for Positive Integer Powers

Find the binomial expansion of \({{\left( 2x+\mathrm{ }5 \right)}^{4}}\), simplifying all the terms.

Hence show that \({{\left( 2x+5 \right)}^{4}}-{{\left( 2x-5 \right)}^{4}}\) can be written as \(320{{x}^{3}}+kx\), where the value of the constant \(k\) is to be stated.

Verify that \(x=2\) is a root of the equation, \({{\left( 2x+5 \right)}^{4}}-{{\left( 2x-5 \right)}^{4}}=3680x-800\) and find the other possible values of \(x\).