1. **State the problem:** We are given the quadratic expression $$4x^2 + 28x + 49$$ and need to factor it or recognize its form. 2. **Recall the formula for factoring perfect square trinomials:** A perfect square trinomial has the form $$a^2 + 2ab + b^2 = (a + b)^2$$ or $$a^2 - 2ab + b^2 = (a - b)^2$$. 3. **Identify terms:** - The first term $$4x^2$$ is a perfect square since $$4x^2 = (2x)^2$$. - The last term $$49$$ is a perfect square since $$49 = 7^2$$. 4. **Check the middle term:** - The middle term should be $$2ab = 2 \times 2x \times 7 = 28x$$, which matches the given middle term. 5. **Write the factorization:** $$4x^2 + 28x + 49 = (2x + 7)^2$$. 6. **Answer:** The expression factors as $$(2x + 7)^2$$. --- For the other expressions: - (7x + 2) feet and (7x - 2) feet are not factorizations of the given quadratic. - (2x + 7) feet matches the factor inside the square. - (2x - 7) feet would correspond to $$4x^2 - 28x + 49$$. For the expression $$x^2 - 16x + ?$$ to be a perfect square trinomial, the missing term is: $$\left(\frac{-16}{2}\right)^2 = (-8)^2 = 64$$. Since 8, 16, and 32 are options, none is correct for a perfect square trinomial with middle term $$-16x$$. **Summary:** - $$4x^2 + 28x + 49 = (2x + 7)^2$$ - Missing term for $$x^2 - 16x + ?$$ to be perfect square is 64 (not listed).