Find all horizontal asymptotes of the following function. \[ f(x)=\frac{5 x^{2}-17 x-12}{10 x+6} \]

To find the horizontal asymptotes of the function \[ f(x)=\frac{5 x^{2}-17 x-12}{10 x+6}, \] we need to analyze the degrees of the polynomial in the numerator and the polynomial in the denominator. The degree of the numerator (which is \(5x^2 - 17x - 12\)) is 2, and the degree of the denominator (which is \(10x + 6\)) is 1. When the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. Instead, it approaches infinity or negative infinity as \(x\) approaches infinity or negative infinity. To confirm this, we can compute the limits as \(x\) approaches infinity: \[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{5 x^{2} - 17 x - 12}{10 x + 6} \] Dividing every term in the numerator and the denominator by \(x\) (the highest power in the denominator): \[ = \lim_{x \to \infty} \frac{5x - 17 - \frac{12}{x}}{10 + \frac{6}{x}}. \] As \(x\) approaches infinity, the terms \(-\frac{17}{x}\) and \(-\frac{12}{x}\) in the numerator go to 0, and the term \(\frac{6}{x}\) in the denominator goes to 0. Thus, we have: \[ \lim_{x \to \infty} f(x) = \frac{5x}{10} = \frac{1}{2} \cdot x \to \infty. \] Therefore, as \(x\) approaches positive infinity, \(f(x)\) approaches infinity, confirming that there is no horizontal asymptote in that direction. Similarly, if we analyze the limit as \(x\) approaches negative infinity: \[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{5 x^{2} - 17 x - 12}{10 x + 6}. \] Following the same steps, we again find: \[ = \lim_{x \to -\infty} \frac{5x}{10} = \frac{1}{2} \cdot x \to -\infty. \] Thus, as \(x\) approaches negative infinity, \(f(x)\) also approaches negative infinity. In conclusion, the function \(f(x)\) has no horizontal asymptotes.