To determine which expression is equivalent to [tex]\(\frac{x^4 - 8x^2 + 16}{x^2 + 5x + 6}\)[/tex], we need to factor both the numerator and the denominator and then simplify the fraction.
1. Factor the numerator:
- The numerator is [tex]\(x^4 - 8x^2 + 16\)[/tex]. We can recognize this as a quadratic in terms of [tex]\(x^2\)[/tex].
- Let's introduce a substitution: [tex]\(u = x^2\)[/tex], so the expression becomes [tex]\(u^2 - 8u + 16\)[/tex].
- Factor the quadratic expression [tex]\(u^2 - 8u + 16\)[/tex]:
[tex]\[
u^2 - 8u + 16 = (u - 4)^2 = (x^2 - 4)^2.
\][/tex]
- Reversing the substitution [tex]\(u = x^2\)[/tex], we get the factored form as [tex]\((x^2 - 4)^2\)[/tex].
- We can further factor [tex]\(x^2 - 4\)[/tex] as a difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2).
\][/tex]
- Therefore, the fully factored form of the numerator is [tex]\((x - 2)^2 (x + 2)^2\)[/tex].
2. Factor the denominator:
- The denominator is [tex]\(x^2 + 5x + 6\)[/tex]. We factor this quadratic expression by finding two numbers that multiply to 6 and add to 5. These numbers are 2 and 3:
[tex]\[
x^2 + 5x + 6 = (x + 2)(x + 3).
\][/tex]
3. Simplify the fraction:
- Now we write the fraction with the factored numerator and denominator:
[tex]\[
\frac{(x - 2)^2 (x + 2)^2}{(x + 2)(x + 3)}.
\][/tex]
- We notice that [tex]\((x + 2)\)[/tex] is a common factor in both the numerator and the denominator, so we can cancel it out:
[tex]\[
\frac{(x - 2)^2 (x + 2)}{x + 3}.
\][/tex]
4. Compare to the given expression:
- The simplified expression is [tex]\(\frac{(x - 2)^2 (x + 2)}{x + 3}\)[/tex].
- The given expression was [tex]\(\frac{(x^2 - 4)^2}{(x + 3)^2}\)[/tex].
These do not match exactly since [tex]\(\frac{(x^2 - 4)^2}{(x + 3)^2}\)[/tex] is equivalent to [tex]\(\frac{((x - 2)(x + 2))^2}{(x + 3)^2}\)[/tex], which would be [tex]\(\frac{(x - 2)^2 (x + 2)^2}{(x + 3)^2}\)[/tex], and that form doesn't align exactly with our above simplified fraction.
Therefore, the expression [tex]\(\frac{(x^2 - 4)^2}{(x + 3)^2}\)[/tex] is not equivalent to [tex]\(\frac{x^4 - 8x^2 + 16}{x^2 + 5x + 6}\)[/tex]. The equivalent simplified form of the given fraction is [tex]\(\frac{(x - 2)^2 (x + 2)}{x + 3}\)[/tex], not [tex]\(\frac{(x^2 - 4)^2}{(x + 3)^2}\)[/tex].
