For any two complex numbers $${z_1},{z_2}$$ and any real number a and b.

$$\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........$$

Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$

Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................

Let $$R$$ $$ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove that $$Rf = {4^{2n + 4}}$$