12. \( \frac{2 t^{2}+6 t+4}{4 t^{2}-12 t-16} \)

Let's simplify the expression \( \frac{2 t^{2}+6 t+4}{4 t^{2}-12 t-16} \). Start with the numerator \( 2t^2 + 6t + 4 \). We can factor it: \[ 2(t^2 + 3t + 2) = 2(t + 1)(t + 2) \] Now for the denominator \( 4t^2 - 12t - 16 \). We can factor out a 4 first: \[ 4(t^2 - 3t - 4) \] Next, we will factor the quadratic \( t^2 - 3t - 4 \): \[ t^2 - 3t - 4 = (t - 4)(t + 1) \] So, the denominator becomes: \[ 4(t - 4)(t + 1) \] Now we can rewrite the entire fraction: \[ \frac{2(t + 1)(t + 2)}{4(t - 4)(t + 1)} \] The \( (t + 1) \) terms cancel out, provided \( t \neq -1 \): \[ \frac{2(t + 2)}{4(t - 4)} \] Now simplify further: \[ \frac{2}{4} \cdot \frac{(t + 2)}{(t - 4)} = \frac{1}{2} \cdot \frac{(t + 2)}{(t - 4)} \] This gives us our final simplified expression: \[ \frac{t + 2}{2(t - 4)} \] Remember, this simplification holds true as long as \( t \neq -1 \) and \( t \neq 4 \) (to avoid division by zero).