Question

1. $d(x) = \frac{x^2 + 3x - 28}{x^2 - 2x - 99}$

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 + 3x - 28\): Find two numbers that multiply to \(-28\) and add to \(3\). They are \(7\) and \(-4\). So \(x^2 + 3x - 28=(x + 7)(x - 4)\).

Factor \(x^2 - 2x - 99\): Find two numbers that multiply to \(-99\) and add to \(-2\). They are \(-11\) and \(9\). So \(x^2 - 2x - 99=(x - 11)(x + 9)\).

Now \(d(x)=\frac{(x + 7)(x - 4)}{(x - 11)(x + 9)}\).

Step2: Identify domain restrictions (if needed) or simplify

To find the domain, set denominator \((x - 11)(x + 9)
eq0\), so \(x
eq11\) and \(x
eq - 9\). The function can't be simplified further as there are no common factors in numerator and denominator.

If the question was to simplify or find domain, this is the process.

Answer:

Simplified form: \(\boldsymbol{\frac{(x + 7)(x - 4)}{(x - 11)(x + 9)}}\) (Domain: \(x
eq11,x
eq - 9\))