Ahmed found the product of (23g2−6n)2%0D%0A(%0D%0A2%0D%0A3%0D%0A%0D%0A2%0D%0A−%0D%0A6%0D%0A%0D%0A)%0D%0A2%0D%0A using the area model. His partial work is shown. What is the partial product that goes in the orange box? What is the partial product that goes in the blue box? What is the product of (23g2−6n)2%0D%0A(%0D%0A2%0D%0A3%0D%0A%0D%0A2%0D%0A−%0D%0A6%0D%0A%0D%0A)%0D%0A2%0D%0A written in standard form?(3 points)

Answer 1

To find the product of \( (23g^2 - 6n)^2 \) and \( (2 - 6) \) using the area model, we need to identify the components of the expressions mentioned.

The expression \( (23g^2 - 6n)^2 \) can be expanded using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):

1. Calculate \( a^2 \) where \( a = 23g^2 \): \[ a^2 = (23g^2)^2 = 529g^4 \]

2. Calculate \( b^2 \) where \( b = 6n \): \[ b^2 = (6n)^2 = 36n^2 \]

3. Calculate \( 2ab \): \[ 2ab = 2 \cdot 23g^2 \cdot 6n = 276g^2n \]

So, \[ (23g^2 - 6n)^2 = 529g^4 - 276g^2n + 36n^2 \]

Next, for \( (2 - 6) \), since \( 2 - 6 \) simplifies to \(-4\), we can find the full expression:

Now, we will multiply this by \(-4\): \[ (23g^2 - 6n)^2 \cdot -4 = (-4)(529g^4 - 276g^2n + 36n^2) \]

Distributing \(-4\) gives:

1. For the term \( 529g^4 \): \[ -4 \cdot 529g^4 = -2116g^4 \]

2. For the term \( -276g^2n \): \[ -4 \cdot (-276g^2n) = 1104g^2n \]

3. For the term \( 36n^2 \): \[ -4 \cdot 36n^2 = -144n^2 \]

So, the final product in standard form is: \[ -2116g^4 + 1104g^2n - 144n^2 \]

Therefore, the partial products are:

• Partial product in the orange box: This will be \(-4 \cdot 529g^4 = -2116g^4\)
• Partial product in the blue box: This will be \(-4 \cdot (-276g^2n) = 1104g^2n\)

And the final product written in standard form is: \[ -2116g^4 + 1104g^2n - 144n^2 \]