1. The problem is to analyze and understand the function $$y = 5 \sin^2\left((x^4 - x + 3)^2\right)$$. 2. The function involves a sine function raised to the power 2, and the argument of sine is itself a square of a polynomial expression. 3. Recall the identity for sine squared: $$\sin^2(\theta) = \left(\sin(\theta)\right)^2$$. Here, $$\theta = (x^4 - x + 3)^2$$. 4. To understand the behavior, note that $$\sin^2(\theta)$$ ranges between 0 and 1 for all real $$\theta$$. 5. Since the sine argument is squared, $$\theta = (x^4 - x + 3)^2 \geq 0$$ for all real $$x$$. 6. The function $$y$$ is scaled by 5, so $$y$$ ranges between 0 and 5. 7. The function is continuous and oscillatory due to the sine squared term, but the frequency and shape depend on the polynomial inside. 8. The polynomial $$x^4 - x + 3$$ grows large for large $$|x|$$, so the sine argument grows rapidly, causing rapid oscillations. 9. Final answer: $$y = 5 \sin^2\left((x^4 - x + 3)^2\right)$$ oscillates between 0 and 5 with increasing frequency as $$|x|$$ increases.