Question

for which values is this expression undefined?\\(\frac{x - 1}{x^2 - 2x - 3} + \frac{5}{2x^2 + 2x}\\)\\(\square\\ x = 0\\)\\(\square\\ x = 5\\)\\(\square\\ x = -2\\)\\(\square\\ x = 1\\)\\(\square\\ x = 3\\)\\(\square\\ x = -1\\)

Explanation:

Step1: Factor denominators

First, factor the denominators of each fraction. For the first fraction, \(x^2 - 2x - 3\) factors to \((x - 3)(x + 1)\) (since \(-3\) and \(1\) multiply to \(-3\) and add to \(-2\)). For the second fraction, \(2x^2 + 2x\) factors to \(2x(x + 1)\) (by factoring out \(2x\)).

Step2: Find values that make denominators zero

A rational expression is undefined when its denominator is zero. So we set each denominator equal to zero and solve for \(x\).

For the first denominator: \((x - 3)(x + 1)=0\). Using the zero - product property, \(x - 3 = 0\) gives \(x = 3\), and \(x+1 = 0\) gives \(x=-1\).

For the second denominator: \(2x(x + 1)=0\). Using the zero - product property, \(2x = 0\) gives \(x = 0\), and \(x + 1=0\) gives \(x=-1\).

So the values of \(x\) that make the expression undefined are \(x = 0\), \(x = 3\), and \(x=-1\).

Answer:

\(x = 0\), \(x = 3\), \(x=-1\) (corresponding to the options: \(x = 0\), \(x = 3\), \(x=-1\))