Select the correct answer. If no denominator equals zero, which expression is equivalent to \( \frac{x+2}{x^{2}-10} \div \frac{2 x+4}{x^{2}+3 x-4} ? \) O. \( \frac{\pi-1}{2(x+4)} \) O. \( \frac{\frac{\pi-1}{2(x+1)}}{\text { B. } \frac{x-1}{2(\pi-4)}} \) O. \( \frac{(x+2)^{2}}{2(x+4)^{1}} \)

To solve the expression \( \frac{x+2}{x^{2}-10} \div \frac{2 x+4}{x^{2}+3 x-4} \), we will first rewrite the division as multiplication by the reciprocal: \[ \frac{x+2}{x^{2}-10} \div \frac{2 x+4}{x^{2}+3 x-4} = \frac{x+2}{x^{2}-10} \cdot \frac{x^{2}+3 x-4}{2 x+4} \] Next, we will simplify each part of the expression. 1. **Factor the denominators and numerators where possible:** - The denominator \( x^{2}-10 \) can be factored as \( (x-\sqrt{10})(x+\sqrt{10}) \). - The numerator \( 2x + 4 \) can be factored as \( 2(x + 2) \). - The quadratic \( x^{2}+3x-4 \) can be factored. We need to find two numbers that multiply to \(-4\) and add to \(3\). These numbers are \(4\) and \(-1\), so we can factor it as \( (x + 4)(x - 1) \). Now substituting these factorizations into the expression gives us: \[ \frac{x+2}{(x-\sqrt{10})(x+\sqrt{10})} \cdot \frac{(x + 4)(x - 1)}{2(x + 2)} \] 2. **Cancel out common factors:** The \( x + 2 \) in the numerator and denominator cancels out: \[ \frac{1}{(x-\sqrt{10})(x+\sqrt{10})} \cdot \frac{(x + 4)(x - 1)}{2} \] This simplifies to: \[ \frac{(x + 4)(x - 1)}{2(x-\sqrt{10})(x+\sqrt{10})} \] 3. **Final expression:** Now we have: \[ \frac{(x + 4)(x - 1)}{2(x^{2}-10)} \] Now, let's compare this with the provided options: - **Option A:** \( \frac{\pi-1}{2(x+4)} \) - **Option B:** \( \frac{\frac{\pi-1}{2(x+1)}}{\text { B. } \frac{x-1}{2(\pi-4)}} \) - **Option C:** \( \frac{(x+2)^{2}}{2(x+4)^{1}} \) None of the options seem to match the simplified expression we derived. However, if we analyze the structure of the options, it appears that none of them are equivalent to the expression we simplified. Thus, the answer is that none of the provided options are equivalent to the original expression.