find all horizontal and vertical asymptotes (if any). (if an answer does not exist, enter dne. enter your…

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to 0. Solve $2x^{2}+5x - 3=0$. Factor the quadratic: $2x^{2}+5x - 3=(2x - 1)(x + 3)=0$. Then $2x-1 = 0$ gives $x=\frac{1}{2}$ and $x + 3=0$ gives $x=-3$.

Step2: Find horizontal asymptotes

Since the degree of the numerator $n = 2$ and the degree of the denominator $m = 2$ are equal. The horizontal - asymptote is $y=\frac{a_{n}}{b_{m}}$, where $a_{n}$ is the leading coefficient of the numerator and $b_{m}$ is the leading coefficient of the denominator. Here $a_{n}=4$ and $b_{m}=2$, so $y=\frac{4}{2}=2$.

Answer:

$x=-3,x=\frac{1}{2},y = 2$