College

Which expression is equivalent to [tex]\left(x^2+11\right)^2+(x-5)(x+5)\,?[/tex]

A. [tex]x^4+23x^2-14[/tex]
B. [tex]x^4+23x^2+96[/tex]
C. [tex]x^4+12x^2+121[/tex]
D. [tex]x^4+x^2+146[/tex]

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).