Write an equation of the slant or nonlinear asymptote. \( m(x)=\frac{36 x^{4}+20 x^{2}-15}{4 x^{2}+4} \) The \( \begin{array}{l}\text { (Choose one) } \\ \text { nonlinear } \\ \text { slant }\end{array} \)

To find the slant or nonlinear asymptote of the function \( m(x)=\frac{36 x^{4}+20 x^{2}-15}{4 x^{2}+4} \), you need to perform polynomial long division because the degree of the numerator is greater than the degree of the denominator. Dividing, you'll find that \( m(x) \) can be approximated by \( 9x^{2} + \frac{20x^{2} - 15 - 36x^{4}}{4x^{2} + 4} \). As \( x \) tends to infinity, the term \(\frac{20x^{2} - 15 - 36x^{4}}{4x^{2} + 4}\) approaches zero, leading to the slant asymptote equation \( y = 9x^{2} \). To summarize, the slant asymptote for the given function is \( y = 9x^{2} \). This means that as the value of \( x \) increases or decreases without bound, the graph of \( m(x) \) will approach the curve described by \( y = 9x^{2} \). So, when you plot the function, keep an eye on how closely it hugs that parabola at both ends!