1) Simplify the pollowing fractions (assume all denomina- tors are non-zero): e, \( \frac{(x+1)\left(x^{2}-36\right)}{x^{2}+8 x+12} \)

To simplify \( \frac{(x+1)(x^{2}-36)}{x^{2}+8x+12} \), we first factor the numerator and the denominator. The numerator can be factored as \( (x+1)(x-6)(x+6) \) since \( x^2 - 36 \) is a difference of squares. The denominator factors to \( (x+2)(x+6) \). Now the expression becomes: \[ \frac{(x+1)(x-6)(x+6)}{(x+2)(x+6)} \] We see that \( (x+6) \) cancels out, leading us to the simplified form: \[ \frac{(x+1)(x-6)}{x+2} \] This is the simplified fraction!