Answer :
Let's simplify the expression [tex]\((x^2 + 4)^2 + (x - 2)(x + 2)\)[/tex] step-by-step to find the equivalent expression.
1. Expand [tex]\((x^2 + 4)^2\)[/tex]:
Using the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], where [tex]\(a = x^2\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[
(x^2 + 4)^2 = (x^2)^2 + 2(x^2)(4) + 4^2
\][/tex]
[tex]\[
= x^4 + 8x^2 + 16
\][/tex]
2. Expand [tex]\((x - 2)(x + 2)\)[/tex]:
This is a difference of squares, which follows the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
(x - 2)(x + 2) = x^2 - 2^2
\][/tex]
[tex]\[
= x^2 - 4
\][/tex]
3. Combine the expanded expressions:
Add the results from step 1 and step 2:
[tex]\[
(x^4 + 8x^2 + 16) + (x^2 - 4)
\][/tex]
[tex]\[
= x^4 + 8x^2 + x^2 + 16 - 4
\][/tex]
[tex]\[
= x^4 + 9x^2 + 12
\][/tex]
Therefore, the expression [tex]\((x^2 + 4)^2 + (x - 2)(x + 2)\)[/tex] simplifies to [tex]\(x^4 + 9x^2 + 12\)[/tex].
The correct answer is D) [tex]\(x^4 + 9x^2 + 12\)[/tex].
1. Expand [tex]\((x^2 + 4)^2\)[/tex]:
Using the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], where [tex]\(a = x^2\)[/tex] and [tex]\(b = 4\)[/tex]:
[tex]\[
(x^2 + 4)^2 = (x^2)^2 + 2(x^2)(4) + 4^2
\][/tex]
[tex]\[
= x^4 + 8x^2 + 16
\][/tex]
2. Expand [tex]\((x - 2)(x + 2)\)[/tex]:
This is a difference of squares, which follows the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
(x - 2)(x + 2) = x^2 - 2^2
\][/tex]
[tex]\[
= x^2 - 4
\][/tex]
3. Combine the expanded expressions:
Add the results from step 1 and step 2:
[tex]\[
(x^4 + 8x^2 + 16) + (x^2 - 4)
\][/tex]
[tex]\[
= x^4 + 8x^2 + x^2 + 16 - 4
\][/tex]
[tex]\[
= x^4 + 9x^2 + 12
\][/tex]
Therefore, the expression [tex]\((x^2 + 4)^2 + (x - 2)(x + 2)\)[/tex] simplifies to [tex]\(x^4 + 9x^2 + 12\)[/tex].
The correct answer is D) [tex]\(x^4 + 9x^2 + 12\)[/tex].
