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Answer :
We start with the expression
[tex]$$
\left(x^2 + 11\right)^2 + (x-5)(x+5).
$$[/tex]
Step 1. Expand the first term.
Expand the square:
[tex]$$
\left(x^2 + 11\right)^2 = \left(x^2\right)^2 + 2 \cdot x^2 \cdot 11 + 11^2 = x^4 + 22x^2 + 121.
$$[/tex]
Step 2. Expand the second term.
Notice that [tex]$(x-5)(x+5)$[/tex] is a difference of squares:
[tex]$$
(x-5)(x+5) = x^2 - 25.
$$[/tex]
Step 3. Combine the results.
Add the two expanded expressions:
[tex]$$
x^4 + 22x^2 + 121 + x^2 - 25.
$$[/tex]
Combine like terms:
- The [tex]$x^4$[/tex] term remains as is.
- 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 expression simplifies to:
[tex]$$
x^4 + 23x^2 + 96.
$$[/tex]
This matches option (B).
[tex]$$
\left(x^2 + 11\right)^2 + (x-5)(x+5).
$$[/tex]
Step 1. Expand the first term.
Expand the square:
[tex]$$
\left(x^2 + 11\right)^2 = \left(x^2\right)^2 + 2 \cdot x^2 \cdot 11 + 11^2 = x^4 + 22x^2 + 121.
$$[/tex]
Step 2. Expand the second term.
Notice that [tex]$(x-5)(x+5)$[/tex] is a difference of squares:
[tex]$$
(x-5)(x+5) = x^2 - 25.
$$[/tex]
Step 3. Combine the results.
Add the two expanded expressions:
[tex]$$
x^4 + 22x^2 + 121 + x^2 - 25.
$$[/tex]
Combine like terms:
- The [tex]$x^4$[/tex] term remains as is.
- 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 expression simplifies to:
[tex]$$
x^4 + 23x^2 + 96.
$$[/tex]
This matches option (B).
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