Answer :
We start with the expression
[tex]$$
(4x^3 + 6x - 7) + (3x^3 - 5x^2 - 5x).
$$[/tex]
Step 1. Remove the parentheses
Since the sign before each parenthesis is positive, we can remove them without changing the signs:
[tex]$$
4x^3 + 6x - 7 + 3x^3 - 5x^2 - 5x.
$$[/tex]
Step 2. Group the like terms
Group the terms by the power of [tex]$x$[/tex]:
- For [tex]$x^3$[/tex]: [tex]$$4x^3 + 3x^3$$[/tex]
- For [tex]$x^2$[/tex]: [tex]$$- 5x^2$$[/tex]
- For [tex]$x$[/tex]: [tex]$$6x - 5x$$[/tex]
- Constant: [tex]$$-7$$[/tex]
Step 3. Combine the like terms
1. Cubic terms:
[tex]$$
4x^3 + 3x^3 = 7x^3.
$$[/tex]
2. Quadratic term:
[tex]$$
-5x^2 \quad \text{(only one term)}
$$[/tex]
3. Linear terms:
[tex]$$
6x - 5x = x.
$$[/tex]
4. Constant term:
[tex]$$
-7 \quad \text{(only one term)}
$$[/tex]
Step 4. Write the simplified expression
Putting it all together, we obtain
[tex]$$
7x^3 - 5x^2 + x - 7.
$$[/tex]
Thus, the simplest form of the given expression is
[tex]$$\boxed{7x^3 - 5x^2 + x - 7}.$$[/tex]
This corresponds to option B.
[tex]$$
(4x^3 + 6x - 7) + (3x^3 - 5x^2 - 5x).
$$[/tex]
Step 1. Remove the parentheses
Since the sign before each parenthesis is positive, we can remove them without changing the signs:
[tex]$$
4x^3 + 6x - 7 + 3x^3 - 5x^2 - 5x.
$$[/tex]
Step 2. Group the like terms
Group the terms by the power of [tex]$x$[/tex]:
- For [tex]$x^3$[/tex]: [tex]$$4x^3 + 3x^3$$[/tex]
- For [tex]$x^2$[/tex]: [tex]$$- 5x^2$$[/tex]
- For [tex]$x$[/tex]: [tex]$$6x - 5x$$[/tex]
- Constant: [tex]$$-7$$[/tex]
Step 3. Combine the like terms
1. Cubic terms:
[tex]$$
4x^3 + 3x^3 = 7x^3.
$$[/tex]
2. Quadratic term:
[tex]$$
-5x^2 \quad \text{(only one term)}
$$[/tex]
3. Linear terms:
[tex]$$
6x - 5x = x.
$$[/tex]
4. Constant term:
[tex]$$
-7 \quad \text{(only one term)}
$$[/tex]
Step 4. Write the simplified expression
Putting it all together, we obtain
[tex]$$
7x^3 - 5x^2 + x - 7.
$$[/tex]
Thus, the simplest form of the given expression is
[tex]$$\boxed{7x^3 - 5x^2 + x - 7}.$$[/tex]
This corresponds to option B.
