Find the Missing Terms: Completing the Quadratic Expression

Complete what is missing:

$(2x+?\rparen^2=4x^2+16x+?$

To solve this problem, we'll follow these steps:

• Step 1: Identify the terms in the binomial $(2x + ?)^2$.
• Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ to ascertain missing values.
• Step 3: Compare with given expanded form and determine the missing terms.

Now, let's work through each step:
Step 1: We have $(2x + ?)^2$. Here, $a = 2x$ and the missing term replaces ?. Assume it is $b$.
Step 2: Use the formula $(a + b)^2 = a^2 + 2ab + b^2$. Replace $a$ with $2x$ giving $a^2 = (2x)^2 = 4x^2$.
Step 3: Comparison with expanded result: We need $2ab$ and $b^2$ such that $(4x^2 + 2ab + b^2 = 4x^2 + 16x + ?$). For $2ab = 16x$, $2(2x)(b) = 16x$ implies $b = 4$. Substituting $b = 4$, we calculate: $b^2 = 4^2 = 16$.

Therefore, the solution to the problem is the missing numbers are 4, 16.