Question

1. \\(\frac{x^2 + 7x + 10}{x + 5} \cdot \frac{x + 1}{x^2 + 3x + 2}\\)

Explanation:

Step1: Factor the quadratics

Factor \(x^2 + 7x + 10\) as \((x + 2)(x + 5)\), \(x^2 + 3x + 2\) as \((x + 1)(x + 2)\).
The expression becomes \(\frac{(x + 2)(x + 5)}{x + 5} \cdot \frac{x + 1}{(x + 1)(x + 2)}\)

Step2: Cancel common factors

Cancel \(x + 5\) (non - zero, so \(x
eq - 5\)), \(x + 1\) (non - zero, so \(x
eq - 1\)), and \(x + 2\) (non - zero, so \(x
eq - 2\)) from numerator and denominator.
After cancellation, we get \(\frac{(x + 2)(x + 5)}{x + 5}\cdot\frac{x + 1}{(x + 1)(x + 2)}=1\) (for \(x
eq - 5,-1,-2\))

Answer:

\(1\) (with the restrictions \(x
eq - 5,-1,-2\))