1. **State the problem:** Simplify or factor the polynomial expression $$8y^8 + 21y^4 + 13$$. 2. **Identify the structure:** Notice that the polynomial is in terms of powers of $y^4$. Let $x = y^4$, then the expression becomes $$8x^2 + 21x + 13$$. 3. **Factor the quadratic in $x$:** We want to factor $$8x^2 + 21x + 13$$. 4. **Find two numbers that multiply to $8 \times 13 = 104$ and add to $21$:** These numbers are $13$ and $8$. 5. **Rewrite the middle term:** $$8x^2 + 13x + 8x + 13$$. 6. **Group terms:** $$(8x^2 + 13x) + (8x + 13)$$. 7. **Factor each group:** $$x(8x + 13) + 1(8x + 13)$$. 8. **Factor out the common binomial:** $$(8x + 13)(x + 1)$$. 9. **Substitute back $x = y^4$:** $$(8y^4 + 13)(y^4 + 1)$$. **Final answer:** $$8y^8 + 21y^4 + 13 = (8y^4 + 13)(y^4 + 1)$$.