Answer :
We start with the expression
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Our goal is to simplify it step by step.
--------------------------------------------------
Step 1. Combine the First Two Polynomials
Add the first two expressions:
[tex]\[
5x^4 - 9x^3 + 7x - 1 \quad \text{and} \quad -8x^4 + 4x^2 - 3x + 2.
\][/tex]
Group like terms:
- For [tex]$x^4$[/tex]:
[tex]$$
5x^4 + (-8x^4) = -3x^4.
$$[/tex]
- For [tex]$x^3$[/tex]:
[tex]$$
-9x^3 \quad (\text{no matching term from the second polynomial}).
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
0 + 4x^2 = 4x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
7x + (-3x) = 4x.
$$[/tex]
- Constants:
[tex]$$
-1 + 2 = 1.
$$[/tex]
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
--------------------------------------------------
Step 2. Expand the Product
Next, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first polynomial by each term in the second:
1. Multiply [tex]$-4x^3$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
-4x^3 \cdot 2x = -8x^4, \quad -4x^3 \cdot (-7) = 28x^3.
$$[/tex]
2. Multiply [tex]$5x$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
5x \cdot 2x = 10x^2, \quad 5x \cdot (-7) = -35x.
$$[/tex]
3. Multiply [tex]$-1$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
-1 \cdot 2x = -2x, \quad -1 \cdot (-7) = 7.
$$[/tex]
Now, combine like terms:
- The [tex]$x^4$[/tex] term is:
[tex]$$
-8x^4.
$$[/tex]
- The [tex]$x^3$[/tex] term is:
[tex]$$
28x^3.
$$[/tex]
- The [tex]$x^2$[/tex] term is:
[tex]$$
10x^2.
$$[/tex]
- The [tex]$x$[/tex] terms:
[tex]$$
-35x - 2x = -37x.
$$[/tex]
- The constant term is:
[tex]$$
7.
$$[/tex]
Thus, the product simplifies to
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
--------------------------------------------------
Step 3. Subtract the Product from the Sum
Now, subtract the expanded product from the sum obtained in Step 1:
[tex]\[
\left[-3x^4 - 9x^3 + 4x^2 + 4x + 1\right] - \left[-8x^4 + 28x^3 + 10x^2 - 37x + 7\right].
\][/tex]
Distribute the negative sign to the second bracket:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
\][/tex]
Combine like terms:
- For [tex]$x^4$[/tex]:
[tex]$$
-3x^4 + 8x^4 = 5x^4.
$$[/tex]
- For [tex]$x^3$[/tex]:
[tex]$$
-9x^3 - 28x^3 = -37x^3.
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
4x^2 - 10x^2 = -6x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
4x + 37x = 41x.
$$[/tex]
- For the constant term:
[tex]$$
1 - 7 = -6.
$$[/tex]
Thus, the simplified expression becomes
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
--------------------------------------------------
Conclusion
The correct simplified form of the given polynomial expression is
[tex]$$
\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}.
$$[/tex]
This corresponds to option C.
[tex]$$
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Our goal is to simplify it step by step.
--------------------------------------------------
Step 1. Combine the First Two Polynomials
Add the first two expressions:
[tex]\[
5x^4 - 9x^3 + 7x - 1 \quad \text{and} \quad -8x^4 + 4x^2 - 3x + 2.
\][/tex]
Group like terms:
- For [tex]$x^4$[/tex]:
[tex]$$
5x^4 + (-8x^4) = -3x^4.
$$[/tex]
- For [tex]$x^3$[/tex]:
[tex]$$
-9x^3 \quad (\text{no matching term from the second polynomial}).
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
0 + 4x^2 = 4x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
7x + (-3x) = 4x.
$$[/tex]
- Constants:
[tex]$$
-1 + 2 = 1.
$$[/tex]
Thus, the sum is
[tex]$$
-3x^4 - 9x^3 + 4x^2 + 4x + 1.
$$[/tex]
--------------------------------------------------
Step 2. Expand the Product
Next, expand the product
[tex]$$
\left(-4x^3 + 5x - 1\right)(2x - 7).
$$[/tex]
Multiply each term in the first polynomial by each term in the second:
1. Multiply [tex]$-4x^3$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
-4x^3 \cdot 2x = -8x^4, \quad -4x^3 \cdot (-7) = 28x^3.
$$[/tex]
2. Multiply [tex]$5x$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
5x \cdot 2x = 10x^2, \quad 5x \cdot (-7) = -35x.
$$[/tex]
3. Multiply [tex]$-1$[/tex] by [tex]$(2x - 7)$[/tex]:
[tex]$$
-1 \cdot 2x = -2x, \quad -1 \cdot (-7) = 7.
$$[/tex]
Now, combine like terms:
- The [tex]$x^4$[/tex] term is:
[tex]$$
-8x^4.
$$[/tex]
- The [tex]$x^3$[/tex] term is:
[tex]$$
28x^3.
$$[/tex]
- The [tex]$x^2$[/tex] term is:
[tex]$$
10x^2.
$$[/tex]
- The [tex]$x$[/tex] terms:
[tex]$$
-35x - 2x = -37x.
$$[/tex]
- The constant term is:
[tex]$$
7.
$$[/tex]
Thus, the product simplifies to
[tex]$$
-8x^4 + 28x^3 + 10x^2 - 37x + 7.
$$[/tex]
--------------------------------------------------
Step 3. Subtract the Product from the Sum
Now, subtract the expanded product from the sum obtained in Step 1:
[tex]\[
\left[-3x^4 - 9x^3 + 4x^2 + 4x + 1\right] - \left[-8x^4 + 28x^3 + 10x^2 - 37x + 7\right].
\][/tex]
Distribute the negative sign to the second bracket:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7.
\][/tex]
Combine like terms:
- For [tex]$x^4$[/tex]:
[tex]$$
-3x^4 + 8x^4 = 5x^4.
$$[/tex]
- For [tex]$x^3$[/tex]:
[tex]$$
-9x^3 - 28x^3 = -37x^3.
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
4x^2 - 10x^2 = -6x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
4x + 37x = 41x.
$$[/tex]
- For the constant term:
[tex]$$
1 - 7 = -6.
$$[/tex]
Thus, the simplified expression becomes
[tex]$$
5x^4 - 37x^3 - 6x^2 + 41x - 6.
$$[/tex]
--------------------------------------------------
Conclusion
The correct simplified form of the given polynomial expression is
[tex]$$
\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}.
$$[/tex]
This corresponds to option C.
