Question

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Explanation:

Step1: Let the unknown polynomial be \( A \). So the equation is \( (x^2 + 12x + 6)-A = 4x^2 + 7x + 1 \). We need to solve for \( A \).

From the equation, we can rearrange it to \( A=(x^2 + 12x + 6)-(4x^2 + 7x + 1) \).

Step2: Expand the right - hand side by distributing the subtraction.

\( A=x^2 + 12x + 6-4x^2-7x - 1 \)

Step3: Combine like terms.

For the \( x^2 \) terms: \( x^2-4x^2=-3x^2 \)
For the \( x \) terms: \( 12x - 7x = 5x \)
For the constant terms: \( 6-1 = 5 \)
So \( A=-3x^2 + 5x+5 \)

For the parts - whole model:

• The top rectangle (the whole before subtraction) is \( x^2 + 12x + 6 \)
• One of the bottom rectangles (the result after subtraction) is \( 4x^2 + 7x + 1 \)
• The other bottom rectangle (the subtrahend) is \( - 3x^2+5x + 5 \) (or we can think of the relationship as whole - part1=part2, where whole is \( x^2 + 12x + 6 \), part1 is \( -3x^2 + 5x+5 \), and part2 is \( 4x^2 + 7x + 1 \))

Answer:

The unknown polynomial is \( -3x^2 + 5x + 5 \). For the parts - whole model:

• Top rectangle: \( x^2 + 12x + 6 \)
• Bottom left rectangle: \( -3x^2 + 5x + 5 \)
• Bottom right rectangle: \( 4x^2 + 7x + 1 \) (or vice - versa for the bottom two rectangles depending on the interpretation of the model, but the polynomial to fill in the blank is \( -3x^2 + 5x + 5 \))