Answer :
We start with the expression
[tex]$$
\left(5x^4 + 13x^2 - x + 10\right) - \left(-6x^4 + 9x^3 + 7\right).
$$[/tex]
The subtraction sign in front of the second polynomial changes the signs of each of its terms. Thus, we have
[tex]$$
5x^4 + 13x^2 - x + 10 + 6x^4 - 9x^3 - 7.
$$[/tex]
Next, we combine like terms:
1. For the [tex]$x^4$[/tex] terms:
[tex]$$
5x^4 + 6x^4 = 11x^4.
$$[/tex]
2. For the [tex]$x^3$[/tex] term, there is only one term:
[tex]$$
-9x^3.
$$[/tex]
3. For the [tex]$x^2$[/tex] term:
[tex]$$
13x^2.
$$[/tex]
4. For the [tex]$x$[/tex] term:
[tex]$$
-x.
$$[/tex]
5. For the constant terms:
[tex]$$
10 - 7 = 3.
$$[/tex]
Putting all these together, the resulting polynomial is
[tex]$$
11x^4 - 9x^3 + 13x^2 - x + 3.
$$[/tex]
Thus, the correct answer is Option 4.
[tex]$$
\left(5x^4 + 13x^2 - x + 10\right) - \left(-6x^4 + 9x^3 + 7\right).
$$[/tex]
The subtraction sign in front of the second polynomial changes the signs of each of its terms. Thus, we have
[tex]$$
5x^4 + 13x^2 - x + 10 + 6x^4 - 9x^3 - 7.
$$[/tex]
Next, we combine like terms:
1. For the [tex]$x^4$[/tex] terms:
[tex]$$
5x^4 + 6x^4 = 11x^4.
$$[/tex]
2. For the [tex]$x^3$[/tex] term, there is only one term:
[tex]$$
-9x^3.
$$[/tex]
3. For the [tex]$x^2$[/tex] term:
[tex]$$
13x^2.
$$[/tex]
4. For the [tex]$x$[/tex] term:
[tex]$$
-x.
$$[/tex]
5. For the constant terms:
[tex]$$
10 - 7 = 3.
$$[/tex]
Putting all these together, the resulting polynomial is
[tex]$$
11x^4 - 9x^3 + 13x^2 - x + 3.
$$[/tex]
Thus, the correct answer is Option 4.
