The number of real solutions of x ^ 7 + 5 x ^ 3 + 3x + 1 = 0 is...

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

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

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

$∴ f\left(x\right)$ 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.

$∴$ number of real solution will be $1$