For real $x$, let $f(x)=x^3+5x+1$, then:

1. $f^{\prime }(x)=3x^2+5$. Since $x^2\ge 0$, $3x^2+5>0$ for all $x\in \mathbb{R}$.As the derivative is strictly positive, the function is strictly increasing, thus $f$ is one-one.2. For an odd degree polynomial, the range is $(-\infty ,\infty )$.As $x\to \infty ,f(x)\to \infty$ and as $x\to -\infty ,f(x)\to -\infty$.Thus, $f$ is onto.The function is a bijection.