1. **State the problem:** Expand and fully simplify the expression $$(h^2 + 2)(h^2 + 3)(h^2 + 5)$$. 2. **Formula and approach:** To expand the product of three binomials, first multiply two of them, then multiply the result by the third. 3. **Step 1: Multiply the first two binomials:** $$ (h^2 + 2)(h^2 + 3) = h^2 \cdot h^2 + h^2 \cdot 3 + 2 \cdot h^2 + 2 \cdot 3 = h^4 + 3h^2 + 2h^2 + 6 = h^4 + 5h^2 + 6 $$ 4. **Step 2: Multiply the result by the third binomial:** $$ (h^4 + 5h^2 + 6)(h^2 + 5) = h^4 \cdot h^2 + h^4 \cdot 5 + 5h^2 \cdot h^2 + 5h^2 \cdot 5 + 6 \cdot h^2 + 6 \cdot 5 $$ 5. **Calculate each term:** $$ = h^6 + 5h^4 + 5h^4 + 25h^2 + 6h^2 + 30 $$ 6. **Combine like terms:** $$ h^6 + (5h^4 + 5h^4) + (25h^2 + 6h^2) + 30 = h^6 + 10h^4 + 31h^2 + 30 $$ 7. **Final answer:** $$ \boxed{h^6 + 10h^4 + 31h^2 + 30} $$ This is the fully expanded and simplified form of the original expression.