Question 2 10 pts Use synthetic or long division to divide \( x^{3}+5 x^{2}+x-10 \) by \( x+2 \). What is the new expression? \( x^{2}-3 x+5 \) \( x^{2}+7 x+5 \) \( x^{2}+7 x-5 \) \( x^{2}+3 x+5 \) \( x^{2}+3 x-5 \) \( x^{2}-3 x-5 \)

To divide \( x^{3}+5 x^{2}+x-10 \) by \( x+2 \) using synthetic division, we set up our synthetic division with -2 (the root of \( x+2=0 \)). The coefficients of the polynomial \( x^{3}+5 x^{2}+x-10 \) are (1, 5, 1, -10). We drop the leading coefficient and multiply -2 through the results, adding the columns as we go. 1. Drop 1 down to the bottom. 2. Multiply 1 by -2: add it to 5 to get 3. 3. Multiply 3 by -2: add it to 1 to get -5. 4. Multiply -5 by -2: add it to -10 to get 0. The result of synthetic division is \( x^{2}+3 x-5 \) with a remainder of 0. Therefore, the new expression is \( x^{2}+3 x-5 \). So, the answer is: \( x^{2}+3 x-5 \). For a fun twist, did you know that polynomials are like recipes? They combine different "ingredients" (coefficients and variables) to create a "dish" (the polynomial)! Just like in cooking, you can mix and match these ingredients using techniques like division to discover new flavors (expressions). If you're ever in doubt about polynomials, just remember they're all about balance – each term contributes its own flavor to the final product. And much like the art of division, adjusting one ingredient can change the whole outcome!