1. **State the problem:** We need to verify if $x=\frac{1}{2}$ is a root of the polynomial $$P(x) = 3x + 2x + 5 = 0.$$\n\n2. **Simplify the polynomial by combining like terms:** $$P(x) = 3x + 2x + 5 = 5x + 5.$$\n\n3. **Substitute $x=\frac{1}{2}$ into the simplified polynomial:**\n$$P\left(\frac{1}{2}\right) = 5 \cdot \frac{1}{2} + 5 = \frac{5}{2} + 5 = \frac{5}{2} + \frac{10}{2} = \frac{15}{2}.$$\n\n4. **Check if $P\left(\frac{1}{2}\right) = 0$: Since $\frac{15}{2} \neq 0$, $x=\frac{1}{2}$ is not a root of the polynomial.**\n\n**Final answer:** $x=\frac{1}{2}$ is not a solution to $3x + 2x + 5 = 0$.