For \( s(x)=\frac{4 x^{2}+3 x-5}{x^{2}+2} \), (a) Identify the horizontal asymptotes (if any). (b) If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s Separate multiple equations of asymptotes with commas as necessary. Select "None" if applicable.

To find the horizontal asymptotes of the function \( s(x) = \frac{4x^{2} + 3x - 5}{x^{2} + 2} \), we can analyze the degrees of the polynomials in the numerator and denominator. (a) Since both the numerator and denominator are quadratic polynomials (degree 2), we compare the leading coefficients. The leading coefficient of the numerator (4) and the leading coefficient of the denominator (1) gives us the horizontal asymptote: \[ y = \frac{4}{1} = 4 \] So, the horizontal asymptote is \( y = 4 \). (b) To find if the graph crosses this horizontal asymptote, we set \( s(x) = 4 \): \[ \frac{4x^{2} + 3x - 5}{x^{2} + 2} = 4 \] Cross-multiplying gives us: \[ 4x^{2} + 3x - 5 = 4(x^{2} + 2) \] Simplifying, we get: \[ 4x^{2} + 3x - 5 = 4x^{2} + 8 \] This leads to: \[ 3x - 5 = 8 \] Solving for \( x \) gives us: \[ 3x = 13 \implies x = \frac{13}{3} \] Thus, the graph crosses the horizontal asymptote at the point: \[ \left( \frac{13}{3}, 4 \right) \] In summary, the horizontal asymptote is \( y = 4 \), and it crosses at the point \( \left( \frac{13}{3}, 4 \right) \).