Answer :
We begin by rewriting the expression in a way that makes it easier to simplify. Notice that the expression can be interpreted as the sum of two parts:
[tex]$$
(x^2 + 11)^2 \quad \text{and} \quad (x-5)(x+5).
$$[/tex]
Step 1. Expand the first term, [tex]$(x^2+11)^2$[/tex]:
Using the formula for a square of a binomial,
[tex]$$
(a+b)^2 = a^2 + 2ab + b^2,
$$[/tex]
where here [tex]$a=x^2$[/tex] and [tex]$b=11$[/tex], we have
[tex]$$
(x^2+11)^2 = (x^2)^2 + 2\cdot(x^2)(11) + 11^2.
$$[/tex]
Calculating each term:
- [tex]$(x^2)^2$[/tex] is [tex]$x^4$[/tex].
- [tex]$2 \cdot (x^2) \cdot 11$[/tex] is [tex]$22x^2$[/tex].
- [tex]$11^2$[/tex] is [tex]$121$[/tex].
So,
[tex]$$
(x^2+11)^2 = x^4 + 22x^2 + 121.
$$[/tex]
Step 2. Expand the second term, [tex]$(x-5)(x+5)$[/tex]:
Recognize that this is a difference of squares which can be written as
[tex]$$
(x-5)(x+5) = x^2 - 5^2.
$$[/tex]
Since [tex]$5^2 = 25$[/tex], we have
[tex]$$
(x-5)(x+5) = x^2 - 25.
$$[/tex]
Step 3. Combine the two expressions:
Add the results from Step 1 and Step 2:
[tex]$$
(x^4 + 22x^2 + 121) + (x^2 - 25).
$$[/tex]
Combine like terms:
- The [tex]$x^4$[/tex] term remains as is: [tex]$x^4$[/tex].
- Combine the [tex]$x^2$[/tex] terms: [tex]$22x^2 + x^2 = 23x^2$[/tex].
- Combine the constant terms: [tex]$121 - 25 = 96$[/tex].
Thus, the combined expression is
[tex]$$
x^4 + 23x^2 + 96.
$$[/tex]
This matches with option B:
[tex]$$
\boxed{x^4+23x^2+96.}
$$[/tex]
[tex]$$
(x^2 + 11)^2 \quad \text{and} \quad (x-5)(x+5).
$$[/tex]
Step 1. Expand the first term, [tex]$(x^2+11)^2$[/tex]:
Using the formula for a square of a binomial,
[tex]$$
(a+b)^2 = a^2 + 2ab + b^2,
$$[/tex]
where here [tex]$a=x^2$[/tex] and [tex]$b=11$[/tex], we have
[tex]$$
(x^2+11)^2 = (x^2)^2 + 2\cdot(x^2)(11) + 11^2.
$$[/tex]
Calculating each term:
- [tex]$(x^2)^2$[/tex] is [tex]$x^4$[/tex].
- [tex]$2 \cdot (x^2) \cdot 11$[/tex] is [tex]$22x^2$[/tex].
- [tex]$11^2$[/tex] is [tex]$121$[/tex].
So,
[tex]$$
(x^2+11)^2 = x^4 + 22x^2 + 121.
$$[/tex]
Step 2. Expand the second term, [tex]$(x-5)(x+5)$[/tex]:
Recognize that this is a difference of squares which can be written as
[tex]$$
(x-5)(x+5) = x^2 - 5^2.
$$[/tex]
Since [tex]$5^2 = 25$[/tex], we have
[tex]$$
(x-5)(x+5) = x^2 - 25.
$$[/tex]
Step 3. Combine the two expressions:
Add the results from Step 1 and Step 2:
[tex]$$
(x^4 + 22x^2 + 121) + (x^2 - 25).
$$[/tex]
Combine like terms:
- The [tex]$x^4$[/tex] term remains as is: [tex]$x^4$[/tex].
- Combine the [tex]$x^2$[/tex] terms: [tex]$22x^2 + x^2 = 23x^2$[/tex].
- Combine the constant terms: [tex]$121 - 25 = 96$[/tex].
Thus, the combined expression is
[tex]$$
x^4 + 23x^2 + 96.
$$[/tex]
This matches with option B:
[tex]$$
\boxed{x^4+23x^2+96.}
$$[/tex]
