Solution: Expand $ (\sin x + 2\cos x)^2 = \sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Add $ \sin^2 x $: total $ 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Simplify using identities: $ 2(1 - \cos^2 x) + 2\sin 2x + 4\cos^2 x = 2 + 2\cos^2 x + 2\sin 2x $. Let $ u = \cos^2 x $, $ \sin 2x = 2\sin x \cos x $. Alternatively, rewrite original expression as $ \sin^2 x + 4\sin x \cos x + 4\cos^2 x + \sin^2 x = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Let $ f(x) = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x $. Use

["Expanding and Simplifying the Trigonometric Expression: A Comprehensive Guide", "Understanding and simplifying complex trigonometric expressions is essential in calculus, physics, engineering, and advanced mathematics. One such expression that frequently arises is:", "$$\n(\sin x + 2\cos x)^2 = \sin^2 x + 4\sin x \cos x + 4\cos^2 x\n$$", "Expanding the square yields a clear form:\n$$\n\sin^2 x + 4\sin x \cos x + 4\cos^2 x\n$$", "Adding an extra $ \sin^2 x $ results in:\n$$\n2\sin^2 x + 4\sin x \cos x + 4\cos^2 x\n$$", "At this stage, mathematical elegance can be restored using fundamental trigonometric identities.", "---", "### Applying Trigonometric Identities", "Recall the Pythagorean identities:\n$$\n\sin^2 x + \cos^2 x = 1 \quad \Rightarrow \quad \sin^2 x = 1 - \cos^2 x \quad \ ext{and} \quad \cos^2 x = 1 - \sin^2 x\n$$", "Substituting $ \sin^2 x = 1 - \cos^2 x $ into the expression:\n$$\n2(1 - \cos^2 x) + 4\sin x \cos x + 4\cos^2 x = 2 - 2\cos^2 x + 4\sin x \cos x + 4\cos^2 x\n$$", "Simplify:\n$$\n2 + 2\cos^2 x + 4\sin x \cos x\n$$", "Now use the double-angle identity:\n$$\n\sin 2x = 2\sin x \cos x \quad \Rightarrow \quad 4\sin x \cos x = 2\sin 2x\n$$", "Also, recall:\n$$\n\cos^2 x = \frac{1 + \cos 2x}{2}\n$$", "Substitute into the expression:\n$$\n2 + 2\left(\frac{1 + \cos 2x}{2}\right) + 2\sin 2x = 2 + (1 + \cos 2x) + 2\sin 2x\n$$", "Finally:\n$$\nf(x) = 3 + \cos 2x + 2\sin 2x\n$$", "---", "### A Powerful Simplified Form", "The expression has now been transformed into a compact form involving double-angle functions:\n$$\n\boxed{f(x) = 3 + \cos 2x + 2\sin 2x}\n$$", "This representation reveals the periodic nature and amplitude-balanced oscillation inherent in the original expression. Instead of working with multiple terms, this unified form is ideal for integration, differentiation, and Fourier analysis.", "---", "### Alternate Path: Using Substitution", "Let’s briefly revisit the expanded form before substituting:\n$$\nf(x) = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x\n$$", "Substitute:\n- $ \sin^2 x = \frac{1 - \cos 2x}{2} $\n- $ \cos^2 x = \frac{1 + \cos 2x}{2} $\n- $ \sin x \cos x = \frac{\sin 2x}{2} $", "Now compute each term:", "1. $ 2\sin^2 x = 2 \cdot \frac{1 - \cos 2x}{2} = 1 - \cos 2x $\n2. $ 4\sin x \cos x = 4 \cdot \frac{\sin 2x}{2} = 2\sin 2x $\n3. $ 4\cos^2 x = 4 \cdot \frac{1 + \cos 2x}{2} = 2 + 2\cos 2x $", "Add them together:\n$$\n(1 - \cos 2x) + 2\sin 2x + (2 + 2\cos 2x) = 3 + \cos 2x + 2\sin 2x\n$$", "Same result confirmed.", "---", "### Why This Expansion Matters", "This transformation not only simplifies the expression but also highlights underlying periodic behavior. Engineers and physicists often convert such forms into amplitude-phase forms for signal analysis:", "$$\n\cos 2x + 2\sin 2x = R\cos(2x - \phi), \quad \ ext{where } R = \sqrt{1^2 + 2^2} = \sqrt{5},\quad \ an\phi = 2\n$$", "This enables efficient harmonic analysis and system modeling.", "---", "### Conclusion", "The journey from expanding $ (\sin x + 2\cos x)^2 $, adding $ \sin^2 x $, and simplifying using trigonometric identities unveils a deeper mathematical structure. By leveraging double-angle identities and substitution, we reduced a quadratic form into a compact linear combination of $ \sin 2x $ and $ \cos 2x $, enhancing both interpretability and computational efficiency.", "Next time you encounter such expressions, expand, apply identities step-by-step, and look for opportunities to transform — transforming complexity into clarity.", "Key Search Terms:\nsimplify $(\sin x + 2\cos x)^2$, trigonometric identity expansion, rewrite $\sin^2 x + 4\sin x \cos x + 4\cos^2 x$, double-angle identity simplification, formula $\sin^2 x + 4\sin x \cos x + 4\cos^2 x$", "---", "This structured approach ensures both mathematical rigor and practical utility, making advanced trigonometric manipulation accessible and efficient."]