Answer: The required simplified form of the given product is
[tex]\dfrac{1}{x^2+2x-35}.[/tex]
Step-by-step explanation: We are given to find the simplified form of the following product :
[tex]P=\dfrac{x+1}{x^2-25}\times\dfrac{x+5}{x^2+8x+7}.[/tex]
To simplify the given product, first we need to factorize the quadratic expressions into linear factors.
The simplification is as follows :
[tex]P\\\\\\=\dfrac{x+1}{x^2-25}\times\dfrac{x+5}{x^2+8x+7}\\\\\\=\dfrac{x+1}{x^2-5^2}\times\dfrac{x+5}{x^2+7x+x+7}\\\\\\=\dfrac{(x+1)}{(x+5)(x-5)}\times\dfrac{(x+5)}{(x+1)(x+7)}\\\\\\=\dfrac{1}{(x-5)(x+7)}\\\\\\=\dfrac{1}{x^2+2x-35}.[/tex]
Thus, the required simplified form of the given product is
[tex]\dfrac{1}{x^2+2x-35}.[/tex]
