Answer:
The option [tex]\frac{(x+5)(x+2)}{x^3-9x}[/tex] is correct
The difference of the given expression is
[tex]\frac{2x+5}{x^2-3x}-(\frac{3x+5}{x^3-9x})-({\frac{x+1}{x^2-9})=\frac{(x+5)(x+2)}{x^3-9x}[/tex]
Step-by-step explanation:
Given expression is [tex]\frac{2x+5}{x^2-3x}-(\frac{3x+5}{x^3-9x})-({\frac{x+1}{x^2-9})[/tex]
To find the difference of the given expression as below :
[tex]\frac{2x+5}{x^2-3x}-(\frac{3x+5}{x^3-9x})-({\frac{x+1}{x^2-9})[/tex]
[tex]=\frac{2x+5}{x(x-3)}-(\frac{3x+5}{x(x^2-9)})-({\frac{x+1}{x^2-9})[/tex]
[tex]=\frac{2x+5}{x(x-3)}-(\frac{3x+5}{x(x^2-3^2)})-({\frac{x+1}{x^2-3^2})[/tex]
[tex]=\frac{2x+5}{x(x-3)}-(\frac{3x+5}{x(x-3)(x+3)})-({\frac{x+1}{(x-3)(x+3)})[/tex]
( using the formula [tex]a^2-b^2=(a+b)(a-b)[/tex] )
[tex]=\frac{2x+5(x+3)-(3x+5)-x(x+1)}{x(x-3)(x+3)}[/tex]
[tex]=\frac{2x^2+6x+5x+15-3x-5-x^2-x}{x(x-3)(x+3)}[/tex] (adding the like terms)
[tex]=\frac{x^2+7x+10}{x(x^2-9)}[/tex] ( by factoring the quadratic polynomial )
[tex]=\frac{(x+5)(x+2)}{x^3-9x}[/tex]
Therefore [tex]\frac{2x+5}{x^2-3x}-(\frac{3x+5}{x^3-9x})-({\frac{x+1}{x^2-9})=\frac{(x+5)(x+2)}{x^3-9x}[/tex]
Therefore the difference of the given expression is
[tex]\frac{(x+5)(x+2)}{x^3-9x}[/tex]
Therefore option [tex]\frac{(x+5)(x+2)}{x^3-9x}[/tex] is correct
