Answer :
We want to simplify the expression
[tex]$$
\left(7x + 2 + 8x^4\right) - \left(2x - 5 - 8x^4\right) + \left(3x + 5x^4\right).
$$[/tex]
Step 1: Distribute the subtraction over the second group.
The term
[tex]$$
-\left(2x - 5 - 8x^4\right)
$$[/tex]
becomes
[tex]$$
-2x + 5 + 8x^4.
$$[/tex]
So, the expression now is:
[tex]$$
7x + 2 + 8x^4 - 2x + 5 + 8x^4 + 3x + 5x^4.
$$[/tex]
Step 2: Group like terms.
Group the [tex]$x^4$[/tex], [tex]$x$[/tex], and constant terms:
- [tex]$x^4$[/tex] terms:
[tex]$$
8x^4 + 8x^4 + 5x^4.
$$[/tex]
- [tex]$x$[/tex] terms:
[tex]$$
7x - 2x + 3x.
$$[/tex]
- Constant terms:
[tex]$$
2 + 5.
$$[/tex]
Step 3: Combine the like terms.
1. For the [tex]$x^4$[/tex] terms:
[tex]$$
8x^4 + 8x^4 + 5x^4 = (8 + 8 + 5)x^4 = 21x^4.
$$[/tex]
2. For the [tex]$x$[/tex] terms:
[tex]$$
7x - 2x + 3x = (7 - 2 + 3)x = 8x.
$$[/tex]
3. For the constant terms:
[tex]$$
2 + 5 = 7.
$$[/tex]
Step 4: Write the simplified expression.
Putting all the combined terms together, we get:
[tex]$$
21x^4 + 8x + 7.
$$[/tex]
Thus, the simplified expression is
[tex]$$
\boxed{21x^4 + 8x + 7}.
$$[/tex]
This corresponds to answer choice D.
[tex]$$
\left(7x + 2 + 8x^4\right) - \left(2x - 5 - 8x^4\right) + \left(3x + 5x^4\right).
$$[/tex]
Step 1: Distribute the subtraction over the second group.
The term
[tex]$$
-\left(2x - 5 - 8x^4\right)
$$[/tex]
becomes
[tex]$$
-2x + 5 + 8x^4.
$$[/tex]
So, the expression now is:
[tex]$$
7x + 2 + 8x^4 - 2x + 5 + 8x^4 + 3x + 5x^4.
$$[/tex]
Step 2: Group like terms.
Group the [tex]$x^4$[/tex], [tex]$x$[/tex], and constant terms:
- [tex]$x^4$[/tex] terms:
[tex]$$
8x^4 + 8x^4 + 5x^4.
$$[/tex]
- [tex]$x$[/tex] terms:
[tex]$$
7x - 2x + 3x.
$$[/tex]
- Constant terms:
[tex]$$
2 + 5.
$$[/tex]
Step 3: Combine the like terms.
1. For the [tex]$x^4$[/tex] terms:
[tex]$$
8x^4 + 8x^4 + 5x^4 = (8 + 8 + 5)x^4 = 21x^4.
$$[/tex]
2. For the [tex]$x$[/tex] terms:
[tex]$$
7x - 2x + 3x = (7 - 2 + 3)x = 8x.
$$[/tex]
3. For the constant terms:
[tex]$$
2 + 5 = 7.
$$[/tex]
Step 4: Write the simplified expression.
Putting all the combined terms together, we get:
[tex]$$
21x^4 + 8x + 7.
$$[/tex]
Thus, the simplified expression is
[tex]$$
\boxed{21x^4 + 8x + 7}.
$$[/tex]
This corresponds to answer choice D.
