For which value(s) of ' a ' , f ( x ) = - x 3 + 4 a x 2 + 2 x - 5 is decreasing for every ' x ' ?

Given that 

$f\left(x\right) = -x^3 + 4a{x}^2 + 2x - 5$

Differentiation on both sides

$f\text{'}\left(x\right) = -3x^2 + 8ax +2$

Given that function is decreasing $\Rightarrow f^{\text{'}}\left(x\right) < 0$

$\Rightarrow -3x^2+8ax+2 <0$

If $A{x}^2+Bx+C < 0 \Rightarrow A < 0, ∆ = B^2 - 4AC <0$

Here Discriminant $∆ = 64a^2 +24$

Which is always positive so for No value of $a$The given function is decreasing.