To simplify the expression \(\frac{x}{x^2 + 28x + 32} + (x + 6)\), we need to combine the terms into a single fraction.
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First, factor \(x^2 + 28x + 32\). We can look for two numbers that multiply to \(32\) and add to \(28\). The numbers \(2\) and \(16\) work, so we can write:
\[ x^2 + 28x + 32 = (x + 2)(x + 16) \]
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Rewrite the expression using this factorization:
\[ \frac{x}{(x + 2)(x + 16)} + (x + 6) \]
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To combine the terms, we want a common denominator, which will be \((x + 2)(x + 16)\). We rewrite \((x + 6)\) with the common denominator:
\[ (x + 6) = \frac{(x + 6) \cdot (x + 2)(x + 16)}{(x + 2)(x + 16)} \]
So we multiply the numerator:
\[ (x + 6) \cdot (x + 2)(x + 16) = (x + 6)(x^2 + 18x + 32) \]
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Now we can express the entire expression with a common denominator:
\[ \frac{x + (x + 6)(x^2 + 18x + 32)}{(x + 2)(x + 16)} \]
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Next, we focus on simplifying the numerator. Expanding \((x + 6)(x^2 + 18x + 32)\):
\[ (x + 6)(x^2 + 18x + 32) = x(x^2 + 18x + 32) + 6(x^2 + 18x + 32) \]
\[ = x^3 + 18x^2 + 32x + 6x^2 + 108x + 192 \]
\[ = x^3 + (18x^2 + 6x^2) + (32x + 108x) + 192 \]
\[ = x^3 + 24x^2 + 140x + 192 \]
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Now include the original term \(x\):
\[ = x + x^3 + 24x^2 + 140x + 192 = x^3 + 24x^2 + 141x + 192 \]
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So, we have:
\[ \frac{x^3 + 24x^2 + 141x + 192}{(x + 2)(x + 16)} \]
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Now, check if the numerator can be factored further. The polynomial \(x^3 + 24x^2 + 141x + 192\) might be factored if we can find a root using the Rational Root Theorem.
Let's test simple integer roots like \(-2\):
\[ (-2)^3 + 24(-2)^2 + 141(-2) + 192 = -8 + 96 - 282 + 192 = -8 + 96 - 282 + 192 = -2 \quad (Not\ a\ root) \]
Testing \( -4 \):
\[ (-4)^3 + 24(-4)^2 + 141(-4) + 192 = -64 + 384 - 564 + 192 = -52 \quad (Not\ a\ root) \]
After checking possible rational roots, you can factor or use polynomial division, but this process often requires trial and error or using a direct factoring method.
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Without finding factors at this stage, we finally arrive at the result:
\[ \frac{x^3 + 24x^2 + 141x + 192}{(x + 2)(x + 16)} \]
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This fraction should be simplified (if possible), but if no simple factors can be established, we keep the answer as:
\[ \frac{x^3 + 24x^2 + 141x + 192}{(x + 2)(x + 16)} \]
The expression cannot be further simplified without specific polynomial roots. Thus, we express it as is.
Final Answer:
\[ \frac{x^3 + 24x^2 + 141x + 192}{(x + 2)(x + 16)} \]