Fill in the Blanks: Solve \((?\times x+?)^2=16x^2+32x+16\) with Completion

Fill in the blanks:

$(?\times x+?)^2=16x^2+32x+16$

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

• Step 1: Identify how to match the quadratic expression with the binomial square.
• Step 2: Compare coefficients for alignment and solve for the unknowns.
• Step 3: Verify by reconstructing the square from known coefficients.

Now, let's work through each step:

Step 1: The given expression is $(?\times x+?)^2 = 16x^2 + 32x + 16$. The goal is to match this with $(ax + b)^2 = a^2x^2 + 2abx + b^2$.

Step 2: Compare the expanded form with the expanded square:

• Match $a^2x^2$ with $16x^2$: $a^2 = 16$. Solving gives $a = 4$.
• Match $2abx$ with $32x$: $2ab = 32$. Substituting $a = 4$, we get $2 \times 4 \times b = 32$ $\Rightarrow b = 4$.
• Match $b^2$ with $16$: $b^2 = 16$. We already confirmed $b = 4$.

Step 3: Since both $a$ and $b$ values match consistently through all comparisons, reconstruct the expression:

$(4x + 4)^2 = (4)^2x^2 + 2(4)(4)x + (4)^2 = 16x^2 + 32x + 16$.

This confirms the correct filling of blanks with consistent polynomial expression alignment.

Therefore, the filled-in expression is $(4 \times x + 4)^2$, matching with the correct choice.

Therefore, the correct solution is $(4, 4)$.