The expression will be undefined if the denominator will be equal to 0.
From the given expression :
[tex]\frac{x-1}{x^2-2x-3}+\frac{5}{2x^2+2x}[/tex]
If any of the denominator equal to 0, the expression will be undefined
Let's find the value of x that will make the first fraction undefined.
The denominator of the first term is :
[tex]x^2-2x-3=0[/tex]
Using factoring :
[tex]\begin{gathered} x^2-2x-3=0 \\ (x+m)(x+n)=0 \end{gathered}[/tex]
We need to think of two numbers, m and n that has a product of -3 and a sum of -2
in this case, m must be -3 and n must be 1.
The sum is -2 and the product is -3
[tex](x-3)(x+1)=0[/tex]
Then find the value of x by equating the factors to 0 :
[tex]\begin{gathered} x-3=0\Rightarrow x=3 \\ x+1=0\Rightarrow x=-1 \end{gathered}[/tex]
So the values of x that will make the first fraction undefined are -1 and 3
Next is to find the values of x to make the 2nd fraction undefined
[tex]\begin{gathered} 2x^2+2x=0 \\ 2x(x+1)=0 \end{gathered}[/tex]
Then equate both factors to 0.
[tex]\begin{gathered} 2x=0\Rightarrow x=0 \\ x+1\Rightarrow x=-1 \end{gathered}[/tex]
The values of x that will make the 2nd fraction undefined are 0 and -1
To summarize :
The values of x that will make the whole expression undefined are :
-1, 3 and 0
