The number of real roots of the equation e ^ 4x + 2 e ^ 3x - e ^ x...

Given that

${e}^{4x} + 2{e}^{3x} - {e}^{x} - 6 = 0$

Let ${e}^{x} = t$, $t > 0$

$f ( t ) = {t}^{4} + 2{t}^{3} - t - 6$
$f ' ( t ) = 4{t}^{3} + 6{t}^{2} - 1$

{f}^{' '} ( t ) = 12{t}^{2} + 12t > 0 $( t > 0 )$

$f ' ( t )$ is increasing function

Now ${f}^{'} ( 0 ) = - 1$

${f}^{'} ( 1 ) = 9$

By using Intermediate value theorem there exists a root for $f\left(x\right) = 0$if $f\left(\alpha\right)f\left(\beta\right) < 0 ⇒ f\left(\gamma\right) = 0, \gamma \in \left(\alpha, \beta\right)$

So ${f}^{'} ( \beta ) = 0$

Hence, only one real root.