1. **State the problem:** Find values of $l$ and $m$ such that the polynomial $$x^4 + 4x^3 + 16x^2 + lx + m$$ is a perfect square. 2. **Assume the polynomial is a perfect square of a quadratic:** Let $$\left(x^2 + ax + b\right)^2 = x^4 + 2ax^3 + (a^2 + 2b)x^2 + 2abx + b^2.$$ 3. **Match coefficients with the given polynomial:** - Coefficient of $x^4$: $1$ matches $1$. - Coefficient of $x^3$: $4 = 2a \implies a = 2$. - Coefficient of $x^2$: $16 = a^2 + 2b = 2^2 + 2b = 4 + 2b \implies 2b = 12 \implies b = 6$. - Coefficient of $x$: $l = 2ab = 2 \times 2 \times 6 = 24$. - Constant term: $m = b^2 = 6^2 = 36$. 4. **Conclusion:** The polynomial is a perfect square when $$l = 24 \quad \text{and} \quad m = 36.$$