1. **State the problem:** Expand and fully simplify the expression $$(x + 5)(x + 3)^2$$. 2. **Recall the formula:** To expand, first recognize that $$(x + 3)^2 = (x + 3)(x + 3)$$. 3. **Expand the square:** $$ (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 $$ 4. **Rewrite the original expression:** $$ (x + 5)(x^2 + 6x + 9) $$ 5. **Distribute each term in $(x + 5)$ across the trinomial:** $$ x(x^2 + 6x + 9) + 5(x^2 + 6x + 9) $$ 6. **Multiply out each term:** $$ x^3 + 6x^2 + 9x + 5x^2 + 30x + 45 $$ 7. **Combine like terms:** $$ x^3 + (6x^2 + 5x^2) + (9x + 30x) + 45 = x^3 + 11x^2 + 39x + 45 $$ 8. **Final answer:** $$ x^3 + 11x^2 + 39x + 45 $$ This is the fully expanded and simplified form of the given expression.