1. Problem (i): Simplify $$\frac{3x+5}{2x^2-9}$$. The denominator can be factored as $$2x^2-9 = 2x^2 - 9 = 2x^2-9$$ is not factorable using integers. So the expression remains $$\frac{3x+5}{2x^2-9}$$. 2. Problem (ii): Simplify $$\frac{\sqrt{x+1}}{\sqrt{x-1}}$$. We combine the radicals: $$\frac{\sqrt{x+1}}{\sqrt{x-1}} = \sqrt{\frac{x+1}{x-1}}$$. 3. Problem (iii): Simplify $$\frac{x^2+1}{x^2+3}$$. No common factors; expression stays $$\frac{x^2+1}{x^2+3}$$. 4. Problem (iv): Simplify $$\frac{1+t+t^2}{1 - t + t^2}$$. This is a ratio of quadratics often left as is; no factorization possible without complex numbers. 5. Problem (vi): Simplify $$\frac{1 - \tan\theta}{1 + \tan\theta}$$. Multiply numerator and denominator by $$\cos\theta$$: $$\frac{1 - \tan\theta}{1 + \tan\theta} = \frac{\cos\theta - \sin\theta}{\cos\theta + \sin\theta}$$. 6. Problem (vii): Simplify $$\frac{\cos x}{1 + \sin^2 x}$$. No further simplification; expression remains as is. 7. Problem (ix): Simplify $$\frac{x \sin x}{1 + \cos x}$$. Using half-angle identity: $$1+\cos x = 2\cos^2 (\frac{x}{2})$$. So, $$\frac{x \sin x}{1 + \cos x} = \frac{x \sin x}{2\cos^2(\frac{x}{2})}$$. 8. Problem (x): Simplify $$\frac{\tan x + \cot x}{3 e^x}$$. Recall $$\tan x = \frac{\sin x}{\cos x}$$ and $$\cot x = \frac{\cos x}{\sin x}$$. So, $$\tan x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}$$. Hence, $$\frac{\tan x + \cot x}{3 e^x} = \frac{1}{3 e^x \sin x \cos x}$$. 9. Problem (xi): Simplify $$\frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta}$$. Multiply numerator and denominator by $$\frac{\cos \theta - \sin \theta}{\cos \theta - \sin \theta}$$: $$= \frac{(\cos \theta - \sin \theta)^2}{\cos^2 \theta - \sin^2 \theta}$$. Numerator expands as $$\cos^2 \theta - 2\sin \theta \cos \theta + \sin^2 \theta = 1 - 2 \sin \theta \cos \theta$$. Denominator is $$\cos 2\theta$$. So expression is $$\frac{1 - 2 \sin \theta \cos \theta}{\cos 2\theta}$$. 10. Problem (xiii): Simplify $$\frac{x^4}{\ln x}$$. Expression remains as is. Final answers: (i) $$\frac{3x+5}{2x^2-9}$$ (ii) $$\sqrt{\frac{x+1}{x-1}}$$ (iii) $$\frac{x^2+1}{x^2+3}$$ (iv) $$\frac{1+t+t^2}{1 - t + t^2}$$ (vi) $$\frac{\cos\theta - \sin\theta}{\cos\theta + \sin\theta}$$ (vii) $$\frac{\cos x}{1 + \sin^2 x}$$ (ix) $$\frac{x \sin x}{2\cos^2(\frac{x}{2})}$$ (x) $$\frac{1}{3 e^x \sin x \cos x}$$ (xi) $$\frac{1 - 2 \sin \theta \cos \theta}{\cos 2\theta}$$ (xiii) $$\frac{x^4}{\ln x}$$