Question: 3 Problem-solving / Reasoning Dan and Emily both expand and simplify the quadratic expression $(x - 5)(-6 + x)$. Dan says the answer is $x^2 - 11x + 30$. Emily says the answer is $x^2 + x - 30$. Only one of them is correct. Who is it? What mistakes were made?
1. **State the problem:**
We need to expand and simplify the quadratic expression $$(x - 5)(-6 + x)$$ and determine which answer, Dan's or Emily's, is correct.
2. **Recall the distributive property (FOIL method):**
$$(a + b)(c + d) = ac + ad + bc + bd$$
We will apply this to $$(x - 5)(-6 + x)$$.
3. **Expand the expression:**
$$x \cdot (-6) + x \cdot x + (-5) \cdot (-6) + (-5) \cdot x$$
which simplifies to
$$-6x + x^2 + 30 - 5x$$
4. **Combine like terms:**
$$x^2 + (-6x - 5x) + 30 = x^2 - 11x + 30$$
5. **Compare with Dan's and Emily's answers:**
- Dan's answer: $$x^2 - 11x + 30$$
- Emily's answer: $$x^2 + x - 30$$
6. **Conclusion:**
Dan's answer is correct.
7. **Identify mistakes:**
- Emily likely made a sign error when multiplying or combining terms, resulting in incorrect coefficients and constant term.
**Final answer:** Dan is correct; Emily made mistakes in sign and combining like terms.