ANSWER
[tex]( \frac{f}{g} )(x ) = \frac{x - 3}{x} \: where \: \: x \ne0, - \frac{1}{2} [/tex]
EXPLANATION
Given:
[tex]f(x) = 2 {x}^{2} - 5x - 3[/tex]
and
[tex]g(x) = 2 {x}^{2} + x[/tex]
Then,
[tex]( \frac{f}{g} )(x ) = \frac{f(x)}{g(x)} [/tex]
Substitute the given functions;
[tex]( \frac{f}{g} )(x ) = \frac{2 {x}^{2} - 5x - 3}{2 {x}^{2} + x} [/tex]
Factor
[tex]( \frac{f}{g} )(x ) = \frac{2 {x}^{2} - 6x + x - 3 }{x(2x + 1)} [/tex]
[tex]( \frac{f}{g} )(x ) = \frac{2x(x - 3) +1 (x - 3)}{x(2x + 1)} [/tex]
[tex]( \frac{f}{g} )(x ) = \frac{(2x + 1)(x - 3) }{x(2x + 1)} [/tex]
Cancel the common factors.
[tex]( \frac{f}{g} )(x ) = \frac{x - 3}{x} \: where \: \: x \ne0, - \frac{1}{2} [/tex]
