$$\eqalign{
& 1)\,\,4{a^2}{b^4} - {c^4}{d^2} \cr
& = {\left( {2a{b^2}} \right)^2} - {\left( {{c^2}d} \right)^2} \cr
& = \left( {2a{b^2} - {c^2}d} \right)\left( {2a{b^2} + {c^2}d} \right) \cr
& 2)\,\,100{x^2} - {\left( {3x - y} \right)^2} \cr
& = {\left( {10x} \right)^2} - {\left( {3x - y} \right)^2} \cr
& = \left( {10x - 3x + y} \right)\left( {10x + 3x - y} \right) \cr
& = \left( {7x + y} \right)\left( {13x - y} \right) \cr
& 3)\,\,{\left( {a + b} \right)^2} - {\left( {2a - b} \right)^2} \cr
& = \left( {a + b - 2a + b} \right)\left( {a + b + 2a - b} \right) \cr
& = 3a\left( {2b - a} \right) \cr
& 4)\,\,{\left( {3x + 1} \right)^2} - 4{\left( {x - 2} \right)^2} \cr
& = {\left( {3x + 1} \right)^2} - {\left[ {2\left( {x - 2} \right)} \right]^2} \cr
& = {\left( {3x + 1} \right)^2} - {\left( {2x - 4} \right)^2} \cr
& = \left( {3x + 1 - 2x + 4} \right)\left( {3x + 1 + 2x - 4} \right) \cr
& = \left( {x + 5} \right)\left( {5x - 3} \right) \cr} $$
