Simplify the expression \( \frac{x^{2}+5 x+6}{x^{2}+7 x+6} \cdot \frac{x^{2}+4 x+3}{x^{2}+6 x+8} \) and give your answer in the form of \( \frac{f(x)}{g(x)} \). Your answer for the function \( f(x) \) is: Your answer for the function \( g(x) \) is:

To simplify the expression \[ \frac{x^{2}+5 x+6}{x^{2}+7 x+6} \cdot \frac{x^{2}+4 x+3}{x^{2}+6 x+8}, \] we can factor each polynomial: 1. \( x^{2}+5x+6 = (x+2)(x+3) \) 2. \( x^{2}+7x+6 = (x+1)(x+6) \) 3. \( x^{2}+4x+3 = (x+1)(x+3) \) 4. \( x^{2}+6x+8 = (x+2)(x+4) \) Now substituting these factors back into the expression gives us: \[ \frac{(x+2)(x+3)}{(x+1)(x+6)} \cdot \frac{(x+1)(x+3)}{(x+2)(x+4)}. \] Next, we can cancel out common factors: - The \( (x+2) \) cancels with \( (x+2) \). - The \( (x+1) \) cancels with \( (x+1) \). - The \( (x+3) \) cancels with \( (x+3) \). We are left with: \[ \frac{1}{(x+4)(x+6)}. \] Thus, we can express this as: \[ \frac{f(x)}{g(x)} = \frac{1}{(x+4)(x+6)}. \] The answer is: Your answer for the function \( f(x) \) is: \( 1 \) Your answer for the function \( g(x) \) is: \( (x+4)(x+6) \)