Answer:
Simplified expression: [tex]\frac{ -20m - 26}{(2m+1)(2m-5)(2m+5)}[/tex]
Restrictions: [tex]m \neq -0.5, m \neq 2.5, m \neq -2.5[/tex]
Step-by-step explanation:
The expression is:
[tex]\frac{2m-1}{4m^2-25} - \frac{2m+5}{4m^2-8m-5}[/tex]
We can simplify the denominator of the first fraction:
[tex]\frac{2m-1}{(2m+5)(2m-5)} - \frac{2m+5}{4m^2-8m-5}[/tex]
Then we can simplify the denominator of the second fraction:
[tex]\frac{2m-1}{(2m+5)(2m-5)} - \frac{2m+5}{(2m+1)(2m-5)}[/tex]
The least common multiple of the denominators is [tex](2m+1)(2m-5)(2m+5)[/tex], therefore we have:
[tex]\frac{(2m-1)(2m+1)}{(2m+1)(2m-5)(2m+5)} - \frac{(2m+5)^2}{(2m+1)(2m-5)(2m+5)}[/tex]
[tex]\frac{4m^2-1}{(2m+1)(2m-5)(2m+5)} - \frac{4m^2+20m+25}{(2m+1)(2m-5)(2m+5)}[/tex]
[tex]\frac{4m^2-1 - 4m^2 - 20m - 25}{(2m+1)(2m-5)(2m+5)}[/tex]
[tex]\frac{ -20m - 26}{(2m+1)(2m-5)(2m+5)}[/tex]
The simplified expression is:
[tex]\frac{ -20m - 26}{(2m+1)(2m-5)(2m+5)}[/tex]
The restrictions are the values of m that makes the denominator zero, so we calculate them using a 'not equal' sign:
[tex](2m+1) \neq 0 \rightarrow m \neq -0.5[/tex]
[tex](2m-5) \neq 0 \rightarrow m \neq 2.5[/tex]
[tex](2m+5) \neq 0 \rightarrow m \neq -2.5[/tex]
