Two dice are rolled and the two numbers are multiplied…

Step 1. Odd products. Both numbers odd: $3 \times 3 = 9$ outcomes. So $36 - 9 = 27$ products are even, and $P = \dfrac{27}{36} = \dfrac{3}{4}$.

Step 2. Perfect squares. The six doubles $(1, 1)$ to $(6, 6)$, and also $(1, 4)$ and $(4, 1)$: $8$ outcomes.

Step 3. The probability. $\dfrac{8}{36} = \dfrac{2}{9}$.