Determine the vertical asymptotes of the graph of the function. Write numbers as simplified fractions or integers. Give your answers in exact form. \[ g(t)=\frac{t-1}{2 t^{2}+3 t-2} \] Part: \( 0 / 3 \) Part 1 of 3 Factor the numerator and denominator. \[ g(t)=\frac{t-1}{\text { [1] }} \]

To find the vertical asymptotes of the function \( g(t) = \frac{t-1}{2t^2 + 3t - 2} \), we first need to factor the denominator. The quadratic in the denominator can be factored by finding two numbers that multiply to \( 2 \times (-2) = -4 \) and add up to \( 3 \). These numbers are \( 4 \) and \( -1 \). Rewriting the quadratic, we can break it down as follows: 1. Rewrite: \[ 2t^2 + 4t - t - 2 \] 2. Group the terms: \[ (2t^2 + 4t) + (-t - 2) \] 3. Factor by grouping: \[ 2t(t + 2) - 1(t + 2) = (2t - 1)(t + 2) \] Now, substituting back into the function, we have: \[ g(t) = \frac{t - 1}{(2t - 1)(t + 2)} \] So the function now looks like: \[ g(t) = \frac{t - 1}{(2t - 1)(t + 2)} \]