22. \( \frac{4 x+1}{5 x+10} \cdot \frac{30 x+60}{2 x-2} \)

First, let's simplify the given expression step-by-step. The original expression is: \[ \frac{4x+1}{5x+10} \cdot \frac{30x+60}{2x-2} \] 1. Factor the numerators and denominators where possible: - The denominator \(5x + 10\) can be factored as \(5(x + 2)\). - The numerator \(30x + 60\) can be factored as \(30(x + 2)\). - The denominator \(2x - 2\) can be factored as \(2(x - 1)\). Now, we rewrite the expression using these factored forms: \[ \frac{4x + 1}{5(x + 2)} \cdot \frac{30(x + 2)}{2(x - 1)} \] 2. Combine the fractions: This gives us: \[ \frac{(4x + 1) \cdot 30(x + 2)}{5(x + 2) \cdot 2(x - 1)} \] 3. Notice that \(x + 2\) appears in both the numerator and denominator, so we can cancel it out (assuming \(x \neq -2\)): \[ \frac{30(4x + 1)}{5 \cdot 2(x - 1)} \] 4. Simplify: Now we can simplify \(30\) in the numerator and constants in the denominator: \[ \frac{30(4x + 1)}{10(x - 1)} = \frac{3(4x + 1)}{(x - 1)} \] Thus, the final simplified expression is: \[ \frac{3(4x + 1)}{(x - 1)} \] Assuming \(x \neq -2\) and \(x \neq 1\) to avoid division by zero.