1. **State the problem:** Factor the quadratic expression $16b^2 - 40b + 25$. 2. **Recall the formula:** A quadratic trinomial $ax^2 + bx + c$ can be factored as $(mx + n)^2$ if it is a perfect square trinomial, where $m^2 = a$, $n^2 = c$, and $2mn = b$. 3. **Check if the trinomial is a perfect square:** - $a = 16$, so $m = 4$ because $4^2 = 16$. - $c = 25$, so $n = 5$ because $5^2 = 25$. - Check $2mn = 2 \times 4 \times 5 = 40$, which matches the middle term coefficient but with a positive sign. 4. **Adjust for the sign:** The middle term is $-40b$, so $n$ should be $-5$. 5. **Write the factorization:** $$16b^2 - 40b + 25 = (4b - 5)^2$$ 6. **Verify by expansion:** $$(4b - 5)^2 = 16b^2 - 2 \times 4b \times 5 + 25 = 16b^2 - 40b + 25$$ **Final answer:** $$\boxed{(4b - 5)^2}$$