Let's simplify the expression step by step. The given expression is:
\[ \frac{5x - 20}{x^2 - 9} \cdot \left( x^2 + 12 + \frac{27}{x^2 + 2x - 24} \right) \]
Step 1: Factor the components
1. Factor \(5x - 20\): \[ 5x - 20 = 5(x - 4) \]
2. Factor \(x^2 - 9\): \[ x^2 - 9 = (x - 3)(x + 3) \quad (\text{difference of squares}) \]
3. Simplify \(x^2 + 12 + \frac{27}{x^2 + 2x - 24}\):
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First, factor \(x^2 + 2x - 24\): \[ x^2 + 2x - 24 = (x + 6)(x - 4) \]
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Now, rewrite \(x^2 + 12\) over the same denominator: \[ x^2 + 12 = \frac{(x^2 + 12)(x + 6)(x - 4)}{(x + 6)(x - 4)} \]
Combining it gives: \[ \frac{(x^2 + 12)(x + 6)(x - 4) + 27}{(x + 6)(x - 4)} \]
Step 2: Combine Fractions
To combine these fractions, we need to fit everything under a common denominator:
\[ x^2 + 12 + \frac{27}{(x + 6)(x - 4)} = \frac{(x^2 + 12)(x + 6)(x - 4) + 27}{(x + 6)(x - 4)} \]
Step 3: Combine Everything
Combine everything into one expression:
\[ \frac{5(x - 4)}{(x - 3)(x + 3)} \cdot \frac{(x^2 + 12)(x + 6)(x - 4) + 27}{(x + 6)(x - 4)} \]
Step 4: Simplify
Looking at our expression, we can cancel \(x - 4\) from the numerator and denominator (note: \(x \neq 4\)):
\[ = \frac{5}{(x - 3)(x + 3)} \cdot \frac{(x^2 + 12)(x + 6) + \frac{27}{(x + 6)}} \]
Step 5: Identify Restrictions
The restrictions on the variable come from the denominators:
- From \(x^2 - 9\) i.e. \(x - 3 \neq 0\) and \(x + 3 \neq 0\), which gives \[ x \neq 3, , x \neq -3 \]
- From \(x^2 + 2x - 24\) i.e. \((x + 6)(x - 4) \neq 0\), which gives \[ x \neq -6, , x \neq 4 \]
Combining these restrictions, we get:
Final Restrictions:
\[ \text{Restrictions: } x \neq 3, -3, -6, 4 \]
Simplified Polynomial in Factored Form:
If \(x^2 + 12 = A\), you may evaluate further depending on expressions specifically about \(A\).
Thus the expression in its simplest factored form becomes:
\[ \frac{5((A)(x + 6) + 27)}{(x - 3)(x + 3)(x + 6)} \]
Simply writing the results:
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Simplified Expression: \[ \frac{5((x^2 + 12)(x + 6) + 27)}{(x - 3)(x + 3)(x + 6)} \]
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Restrictions: \[ -6, -3, 3, 4 \]