What is the quotient of the rational expression shown below? Make sure your answer is in reduced form. X^2+7X+10/X-2÷X^-25/4X-8

Answer:Simplified form of [tex]\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} }   = \frac{4(X+2)}{(X-5)}[/tex]Step-by-step explanation:Here, the given expression is [tex]\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} }   = \frac{P(X)}{Q(X)} \\\implies P(X) = {\frac{X^2+7X+10}{X-2} , Q(X) = \frac{X^2-25}{4X-8}[/tex]Now, using the Algebraic Identities:[tex](a^2 - b^2) = (a+b)(a-b)[/tex]Simplify P(X) and Q(X) separably:[tex]P(X) = \frac{X^2+7X+10}{X-2} =   \frac{X^2+5X + 2X+10}{X-2} \\\implies P(X) = {\frac{X(X + 5) +2(X+5)}{X-2}   \\= P(X) = {\frac{(X+ 5)(X+2)}{X-2} \\\\\implies  P(X) =  {\frac{(X+ 5)(X+2)}{X-2}[/tex]Similarly, Q(X) = [tex]\frac{X^2-25}{4X-8}  = \frac{(X-5)(X+5)}{4X-8} \\\implies Q(X) =  \frac{(X-5)(X+5)}{4(X-2)}[/tex]hence, the given fraction is simplified to [tex]\frac{P(X)}{Q(X)}  = \frac{\frac{(X+ 5)(X+2)}{(X-2)} }{\frac{(X-5)(X+5)}{4(X-2)} }[/tex][tex]\implies \frac{P(X)}{Q(X)}  = \frac{\frac{(X+ 5)(X+2)}{(X-2)} }{\frac{(X-5)(X+5)}{4(X-2)} }  ={\frac{(X+ 5)(X+2)}{(X-2)} } \times  {\frac{ 4(X-2)}{(X-5)(X+5)}   = \frac{4(X+2)}{(X-5)}[/tex]Hence, the simplified form of [tex]\frac{\frac{X^2+7X+10}{X-2} }{\frac{X^2-25}{X-8} }   = \frac{4(X+2)}{(X-5)}[/tex]