Find the set of all $$x$$ for which $${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$$

For $$a \le 0,$$ determine all real roots of the equation $$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$$

Solve for $$x$$ ; $${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$$

If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show that $$${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$$