Inequality -- 4

$4x^2+2x+a+2 \geq 2ax$ $\\$ $4x^2+2x+2 \geq (2x-1)a$ $\\$ $\because x\in [1, \infty)$ $\therefore 2x-1 \geq 1$ $\\$ $\quad$ $\\$ $\therefore a \leq \frac {4x^2+2x+2}{2x-1}$ $\quad$ [The key is to create a equation of a using only variable x] $\\$
$a \leq \frac {(2x-1)^2 + 3(2x-1)+4}{2x-1}$ $\\$ $a \leq (2x-1)+\frac {4}{2x-1} + 3$ $\\$ $a \leq 2 \sqrt 4 + 3$ $\\$ $a \leq 7$