ANSWER
[tex]( \frac{g}{f} )(x) = \frac{1}{ 2x+3} [/tex]
The domain is:
[tex]x \ne1 \: or \: x \ne - \frac{3}{2}[/tex]
EXPLANATION
The given functions are:
[tex]f(x) = 2 {x}^{2} + x - 3[/tex]
and
[tex]g(x) = x - 1[/tex]
[tex]( \frac{g}{f} )(x) = \frac{g(x)}{f(x)} [/tex]
[tex]( \frac{g}{f} )(x) = \frac{x - 1}{2 {x}^{2} + x - 3 } [/tex]
Factor the quadratic trinomial in the denominator.
[tex]( \frac{g}{f} )(x) = \frac{x - 1}{2 {x}^{2} +3 x - 2x- 3 } [/tex]
[tex]( \frac{g}{f} )(x) = \frac{x - 1}{ x(2x+3) -1 (2x + 3 )} [/tex]
[tex]( \frac{g}{f} )(x) = \frac{x - 1}{ (2x+3) (x-1)} [/tex]
Cancel the common factors,
[tex]( \frac{g}{f} )(x) = \frac{1}{ 2x+3} \: where \: x \ne1 \: or \: x \ne- \frac{3}{2} [/tex]
