One more hint for 4.37: $\dfrac{x^3}{1-x^8}=\dfrac{x^4}{x(1-x^8)}$. Apply the method from 4.9 on the denominator.

4.38: Use the following inequalities: \[ kx^2 + \frac{y^2}{4k} \geq xy; \quad y^2+z^2 \geq 2yz; \quad \frac{z^2}{4k} + kw^2 \geq zw. \]