if x=sqrt(3)/(sqrt(2)-1) find 2x^2+10x+12/x^2+3x

To find the value of the given expression \( \frac{2x^2 + 10x + 12}{x^2 + 3x} \) when \( x = \frac{\sqrt{3}}{\sqrt{2} - 1} \), let's follow these steps:

1. **Simplify \( x \)**
   
   We start with \( x = \frac{\sqrt{3}}{\sqrt{2} - 1} \). To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator \( \sqrt{2} + 1 \):

   \[
   x = \frac{\sqrt{3}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{3}(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)}
   \]

   The denominator simplifies as follows:

   \[
   (\sqrt{2} - 1)(\sqrt{2} + 1) = (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1
   \]

   So, \( x \) simplifies to:

   \[
   x = \sqrt{3}(\sqrt{2} + 1) = \sqrt{6} + \sqrt{3}
   \]

2. **Calculate \( x^2 \)**

   \[
   x = \sqrt{6} + \sqrt{3}
   \]

   Now we calculate \( x^2 \):

   \[
   x^2 = (\sqrt{6} + \sqrt{3})^2 = (\sqrt{6})^2 + 2(\sqrt{6})(\sqrt{3}) + (\sqrt{3})^2
   \]

   \[
   x^2 = 6 + 2\sqrt{18} + 3 = 6 + 6\sqrt{2} + 3 = 9 + 6\sqrt{2}
   \]

3. **Evaluate \( 2x^2 + 10x + 12 \)**

   \[
   2x^2 = 2(9 + 6\sqrt{2}) = 18 + 12\sqrt{2}
   \]

   \[
   10x = 10(\sqrt{6} + \sqrt{3}) = 10\sqrt{6} + 10\sqrt{3}
   \]

   So, \( 2x^2 + 10x + 12 \):

   \[
   2x^2 + 10x + 12 = 18 + 12\sqrt{2} + 10\sqrt{6} + 10\sqrt{3} + 12 = 30 + 12\sqrt{2} + 10\sqrt{6} + 10\sqrt{3}
   \]

4. **Evaluate \( x^2 + 3x \)**

   \[
   3x = 3(\sqrt{6} + \sqrt{3}) = 3\sqrt{6} + 3\sqrt{3}
   \]

   So, \( x^2 + 3x \):

   \[
   x^2 + 3x = 9 + 6\sqrt{2} + 3\sqrt{6} + 3\sqrt{3}
   \]

5. **Divide the two expressions**

   \[
   \frac{2x^2 + 10x + 12}{x^2 + 3x} = \frac{30 + 12\sqrt{2} + 10\sqrt{6} + 10\sqrt{3}}{9 + 6\sqrt{2} + 3\sqrt{6} + 3\sqrt{3}}
   \]

Given that both the numerator and the denominator are quite complex, it is non-trivial to simplify this expression further without numeric approximation or additional context. However, at this point, we have derived the required algebraic expressions.

Would you like any further details or have any questions on specific steps?

Here are 5 numbered questions you might want to ask:
1. Can you show how to simplify complex rational expressions further?
2. How do you rationalize the denominator in more complex fractions?
3. Could you help me understand how to simplify expressions involving square roots?
4. What are some techniques to verify the accuracy of algebraic manipulations?
5. Can you explain how to handle quadratic expressions in greater detail?

**Tip:** When dealing with complex algebraic expressions, always simplify the terms step by step, and verify each step to avoid errors.

Learn how to evaluate the complex expression 2x^2 + 10x + 12 divided by x^2 + 3x using the given value of x = sqrt(3) / (sqrt(2) - 1). Follow a detailed step-by-step solution involving algebraic manipulations and rationalizing the denominator. Suitable for university-level mathematics.

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