The number of real solutions of x 7 + 5 x 3 + 3 x + 1 = 0 is equal to _____.

$f\left(x\right)=x^7+5x^3+3x+1$

For number of real solutions, we need to find the number of times the curve cuts $x-$axis.

Now $f^{\text{'}}\left(x\right)=7x^6+15x^2+3>0$

$\therefore f\left(x\right)$ is strictly increasing function

i.e. when $x\to -\infty ,y\to -\infty$ and when  $x\to \infty ,y\to \infty$

So, the curve will cut $x-$axis only once.

$\therefore$ number of real solution will be $1$