1. **State the problem:** Simplify or analyze the expression $180 - 2w^2 + 3w$. 2. **Identify the type of expression:** This is a quadratic expression in terms of $w$ because it contains a $w^2$ term. 3. **Rewrite the expression in standard form:** $$-2w^2 + 3w + 180$$ 4. **Explain the components:** - The coefficient of $w^2$ is $-2$ (which means the parabola opens downward if graphed). - The coefficient of $w$ is $3$. - The constant term is $180$. 5. **If asked to find roots or factor, use the quadratic formula:** $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -2$, $b = 3$, and $c = 180$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4(-2)(180) = 9 + 1440 = 1449$$ 7. **Find the roots:** $$w = \frac{-3 \pm \sqrt{1449}}{2(-2)} = \frac{-3 \pm \sqrt{1449}}{-4}$$ 8. **Interpretation:** Since the discriminant is positive, there are two real roots. **Final answer:** The expression is a quadratic with two real roots given by $$w = \frac{-3 \pm \sqrt{1449}}{-4}$$