Instructions: \( \begin{array}{l}\text { You MUST show all relevant steps to receive full mari } \\ \text { If needed, round your final answer to one decimal places. }\end{array} \) 1. Perform the indicated operation and fully simplify. \[ \frac{5 x}{x^{2}+3 x-10}-\frac{7}{2 x+10} \]
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To perform the operation and simplify the expression \(\frac{5 x}{x^{2} + 3 x - 10} - \frac{7}{2 x + 10}\), we’ll first factor the denominators to make the subtraction easier. **Step 1: Factor the denominators** 1. For the first denominator \(x^{2} + 3 x - 10\), we need to find two numbers that multiply to \(-10\) and add to \(3\). These numbers are \(5\) and \(-2\): \[ x^{2} + 3x - 10 = (x + 5)(x - 2) \] 2. For the second denominator \(2x + 10\), we can factor out \(2\): \[ 2x + 10 = 2(x + 5) \] Now, we rewrite the expression: \[ \frac{5x}{(x + 5)(x - 2)} - \frac{7}{2(x + 5)} \] **Step 2: Find a common denominator** The common denominator for both fractions is \(2(x + 5)(x - 2)\). **Step 3: Rewrite each fraction with the common denominator** 1. For the first fraction: \[ \frac{5x}{(x + 5)(x - 2)} \cdot \frac{2}{2} = \frac{10x}{2(x + 5)(x - 2)} \] 2. For the second fraction: \[ \frac{7}{2(x + 5)} \cdot \frac{(x - 2)}{(x - 2)} = \frac{7(x - 2)}{2(x + 5)(x - 2)} \] Now, the rewritten expression is: \[ \frac{10x - 7(x - 2)}{2(x + 5)(x - 2)} \] **Step 4: Simplify the numerator** Simplifying the numerator \(10x - 7(x - 2)\): \[ 10x - 7x + 14 = 3x + 14 \] So, the expression becomes: \[ \frac{3x + 14}{2(x + 5)(x - 2)} \] **Step 5: Final representation** The final simplified expression is: \[ \frac{3x + 14}{2(x + 5)(x - 2)} \] If needed, we can evaluate this expression for specific values of \(x\) but as it stands, this is fully simplified with all steps shown!