1. **State the problem:** We have a circle with center P and diameters AC and BE. The circle is divided into arcs with measures given as expressions in terms of $w$: $(4w + 8)^\circ$, $(2w + 11)^\circ$, and $(4w + 4)^\circ$. We need to find the measure of the minor arc DE in degrees. 2. **Understand the problem:** The total measure of all arcs around the circle is $360^\circ$. The sum of the given arcs plus the arc DE must equal $360^\circ$. 3. **Write the equation:** $$ (4w + 8) + (2w + 11) + (4w + 4) + \text{arc DE} = 360 $$ 4. **Simplify the sum of known arcs:** $$ (4w + 8) + (2w + 11) + (4w + 4) = 4w + 8 + 2w + 11 + 4w + 4 = (4w + 2w + 4w) + (8 + 11 + 4) = 10w + 23 $$ 5. **Express arc DE:** $$ \text{arc DE} = 360 - (10w + 23) = 360 - 10w - 23 = 337 - 10w $$ 6. **Final answer:** The measure of the minor arc DE is $$ \boxed{337 - 10w}\text{ degrees} $$ This expression gives the arc measure of DE in terms of $w$.