Hello!
[tex]\large\boxed{x + 2}[/tex]
[tex]\frac{2x^{2} +x}{2x-2} + \frac{x-4}{2x-2}[/tex]
The denominators are equal, so we can simply combine the numerator:
[tex]\frac{2x^{2} +x+ x - 4}{2x-2}[/tex]
Combine like terms in the numerator:
[tex]\frac{2x^{2} +2x - 4}{2x-2}[/tex]
Factor out 2 from the entire fraction:
[tex]\frac{2(x^{2} +x - 2)}{2(x-1)}[/tex]
Cancel out 2 from both the numerator and denominator:
[tex]\frac{x^{2} +x - 2}{x-1}[/tex]
Factor the numerator to simplify further:
[tex]\frac{(x + 2)(x - 1)}{x-1}[/tex]
Cancel out x - 1:
[tex]x + 2[/tex]
