# Technical Document Extraction: Mathematical Problem Solution
## Problem Statement
The image contains a step-by-step solution to the equation:
$$
\$441 + 2(21)(19) + 361 = x\$
$$
The goal is to solve for \$x\$.
---
## Step-by-Step Solution
### Step 1: Algebraic Identity Application
**Text**:
"We know that \$a^2 + 2ab + b^2 = (a + b)^2\$. [score = 0.33]"
**Interpretation**:
The solution begins by recognizing the algebraic identity for perfect square trinomials.
---
### Step 2: Substitution
**Text**:
"So \$441 + 2(21)(19) + 361 = (21 + 19)^2\$. [score = 0.49]"
**Interpretation**:
The values \$441\$ and \$361\$ are identified as \$21^2\$ and \$19^2\$, respectively. The middle term \$2(21)(19)\$ matches the identity's \$2ab\$ component.
---
### Step 3: Simplification
**Text**:
"So we get \$(21 + 19)^2 = x\$. [score = 0.56]"
**Interpretation**:
The expression simplifies to \$(40)^2 = x\$.
---
### Step 4: Final Calculation
**Text**:
"So we get \$x = 40^2 = 1600\$. [score = 0.90]"
**Interpretation**:
The value of \$x\$ is calculated as \$1600\$.
---
### Step 5: Verification
**Text**:
"So we get \$x = 1600\$. [score = 0.94]
And we know that \$x = 1600\$. [score = 0.95]
That's our answer. [score = 0.93]"
**Interpretation**:
Multiple verification steps confirm the solution \$x = 1600\$.
---
### Step 6: Final Confirmation
**Text**:
"So \$441 + 2(21)(19) + 361 = 1600\$. [score = 0.93]
So \$x = 1600\$. [score = 0.93]
Right. So \$441 + 2(21)(19) + 361 = 1600\$. [score = 0.94]
#### 1600 [score = 0.91]"
**Interpretation**:
The solution is reiterated with high confidence scores, concluding with the final answer \$x = 1600\$.
---
## Key Observations
1. **Mathematical Identity**: The solution leverages the identity \$(a + b)^2 = a^2 + 2ab + b^2\$.
2. **Substitution**: Values \$21\$ and \$19\$ are substituted into the identity.
3. **Scores**: Each step includes a numerical score (e.g., [score = 0.93]), likely indicating automated evaluation or confidence metrics.
4. **Final Answer**: The value of \$x\$ is consistently identified as \$1600\$.
---
## Conclusion
The equation \$441 + 2(21)(19) + 361 = x\$ is solved by recognizing it as a perfect square trinomial. The solution process involves substitution, simplification, and verification, with scores provided at each step. The final answer is:
$$
x = 1600
$$
