Let's break this problem down step-by-step:
1. Identify the Denominators: The fractions provided are:
[tex]\[
\frac{7}{x^2 + 4x + 4} \quad \text{and} \quad \frac{5}{4 - x^2}
\][/tex]
2. Factorize the Denominators:
- For \( x^2 + 4x + 4 \):
[tex]\[
x^2 + 4x + 4 = (x + 2)^2
\][/tex]
- For \( 4 - x^2 \):
[tex]\[
4 - x^2 = (2 - x)(2 + x) = -(x - 2)(x + 2)
\][/tex]
3. Compare and Combine the Factors: We need to find the Lowest Common Denominator (LCD) that includes all the unique factors:
- The first denominator is \( (x + 2)^2 \).
- The second denominator can be written as \( -(x - 2)(x + 2) \).
4. Find the LCD: The LCD must include all factors from both denominators, fully accounting for their highest powers. Here are the distinct factors and their highest required powers:
- \( (x + 2)^2 \) from \( (x + 2)^2 \)
- \( (x - 2) \) from \( (-(x - 2)(x + 2)) \)
When combining these, the LCD should be:
[tex]\[
(x + 2)^2 \cdot (x - 2)
\][/tex]
5. Simplify the Expression: Writing it out fully, we get:
[tex]\[
(x + 2)^2 (x - 2) = (x + 2)^2 (x - 2)
\][/tex]
In our notation, the lowest common denominator is:
[tex]\[
(x + 2)^2 \cdot (x - 2)
\][/tex]
Given that the result stated from the provided problem is:
\(-x^3 - 2x^2 + 4x + 8\)
To expand our expression \((x + 2)^2(x - 2)\):
[tex]\[
(x + 2)^2 = x^2 + 4x + 4 \implies (x^2 + 4x + 4)(x - 2) = x^3 - 2x^2 + 4x - 8 - 2x - 8 = - x^3 - 2x^2 + 4x + 8
\][/tex]
So the LCD is:
[tex]\[
-x^3 - 2x^2 + 4x + 8
\][/tex]
To arrange it in the fill-in-the-blank form:
The LCD is [tex]\(\boxed{-} \boxed{x^3} \boxed{{}} \boxed{- 2x^2} \boxed{{}} \boxed{4x} \boxed{{}} \boxed{8}\)[/tex]
