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Simplify (8n2+2+6n3)−(2−7n4+n3)(8n2+2+6n3)−(2−7n4+n3) . Fill in the missing coefficient for each term. Coefficients may be used more than once or not at all. Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. Response area n4+n4+ Response area n3+n3+ Response area n2+n2+ Response area n+n+ Response area 7508

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To simplify the expression \((8n^2+2+6n^3) - (2-7n^4+n^3)(8n^2+2+6n^3) - (2-7n^4+n^3)\), we will follow these steps:

  1. Start by simplifying the expression inside the parentheses,
  2. Then perform the multiplication,
  3. Finally, combine the terms correctly.

First, let's denote the first polynomial as \( P(n) = 8n^2 + 2 + 6n^3 \) and the second polynomial as \( Q(n) = 2 - 7n^4 + n^3 \).

Now, we need to calculate \( Q(n) \cdot P(n) \):

\[ Q(n) \cdot P(n) = (2 - 7n^4 + n^3)(8n^2 + 2 + 6n^3) \]

To find this product, we can distribute each term in \( Q(n) \):

  1. \( 2(8n^2 + 2 + 6n^3) = 16n^2 + 4 + 12n^3 \)
  2. \( -7n^4(8n^2 + 2 + 6n^3) = -56n^6 - 14n^4 - 42n^7 \)
  3. \( n^3(8n^2 + 2 + 6n^3) = 8n^5 + 2n^3 + 6n^6 \)

Now combining these products, we get:

\[ Q(n) \cdot P(n) = -42n^7 - 56n^6 + 8n^5 + (12n^3 + 2n^3 - 14n^4 + 4 + 16n^2) \]

Reorganizing all the terms gives:

\[ -42n^7 - 56n^6 + 8n^5 - 14n^4 + 14n^3 + 16n^2 + 4 \]

Next, we need to find the entire simplified expression:

\[ P(n) - Q(n) \cdot P(n) - Q(n) \]

This gives:

  1. Simplifying \( P(n) - Q(n) \): \[ P(n) - Q(n) = (8n^2 + 2 + 6n^3) - (2 - 7n^4 + n^3) \] Which simplifies to: \[ 6n^3 - n^3 + 7n^4 + 8n^2 + 2 - 2 = 7n^4 + 5n^3 + 8n^2 \]

Now:

\[ (7n^4 + 5n^3 + 8n^2) - (-42n^7 - 56n^6 + 8n^5 - 14n^4 + 14n^3 + 16n^2 + 4) \]

The final expression will thus be:

\[ 42n^7 + 56n^6 - 7n^4 - 3n^3 - 8n^2 - 4 \]

Now, filling in the coefficients:

  • For the \( n^4 \) term: coefficient = -7.
  • For the \( n^3 \) term: coefficient = -3.
  • For the \( n^2 \) term: coefficient = -8.
  • For the \( n \) term: coefficient = 0 (not present).
  • Constant term: coefficient = -4.

In summary, the coefficients for each term to fill in the inputs are:

  • Coefficient of \( n^4 \): -7
  • Coefficient of \( n^3 \): -3
  • Coefficient of \( n^2 \): -8
  • Coefficient of \( n \): 0
  • Constant = -4.

Please input these coefficients in the respective places.