To simplify the problem \(\frac{(5x+4)(3x+2)+(x^2-6)(x-2)}{x-2}\), we can follow these steps:
Step 1: Expand the Numerator
First, we need to expand the terms in the numerator, which consists of two parts: \((5x+4)(3x+2)\) and \((x^2-6)(x-2)\).
1. Expand \((5x + 4)(3x + 2)\):
\[ (5x + 4)(3x + 2) = 5x \cdot 3x + 5x \cdot 2 + 4 \cdot 3x + 4 \cdot 2 \] \[ = 15x^2 + 10x + 12x + 8 = 15x^2 + 22x + 8 \]
2. Expand \((x^2 - 6)(x - 2)\):
\[ (x^2 - 6)(x - 2) = x^2 \cdot x + x^2 \cdot (-2) - 6 \cdot x - 6 \cdot (-2) \] \[ = x^3 - 2x^2 - 6x + 12 \]
Step 2: Combine the Expanded Terms
Now, we combine both expanded parts:
\[ 15x^2 + 22x + 8 + (x^3 - 2x^2 - 6x + 12) \]
Combine like terms: \[ x^3 + (15x^2 - 2x^2) + (22x - 6x) + (8 + 12) \] \[ = x^3 + 13x^2 + 16x + 20 \]
Thus, the numerator can be expressed as: \[ x^3 + 13x^2 + 16x + 20 \]
Step 3: Put it All Together
Putting everything together, the expression simplifies to:
\[ \frac{x^3 + 13x^2 + 16x + 20}{x - 2} \]
Step 4: Perform Polynomial Long Division
Now, we need to perform polynomial long division of \(x^3 + 13x^2 + 16x + 20\) by \(x - 2\).
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Divide the leading term: \(x^3 \div x = x^2\).
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Multiply \(x^2\) by \(x - 2\): \[ x^2(x - 2) = x^3 - 2x^2 \]
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Subtract: \[ (x^3 + 13x^2 + 16x + 20) - (x^3 - 2x^2) = 15x^2 + 16x + 20 \]
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Repeat:
- Divide the leading term: \(15x^2 \div x = 15x\).
- Multiply \(15x\) by \(x - 2\): \[ 15x(x - 2) = 15x^2 - 30x \]
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Subtract: \[ (15x^2 + 16x + 20) - (15x^2 - 30x) = 46x + 20 \]
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Repeat:
- Divide the leading term: \(46x \div x = 46\).
- Multiply \(46\) by \(x - 2\): \[ 46(x - 2) = 46x - 92 \]
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Subtract: \[ (46x + 20) - (46x - 92) = 112 \]
Result of Division
The result of the polynomial long division is:
\[ x^2 + 15x + 46 + \frac{112}{x - 2} \]
Final Answer
Thus, the simplified form of the original expression is:
\[ x^2 + 15x + 46 + \frac{112}{x - 2} \]
This is the simplified expression in standard form.