For `(1)/(asqrt(x)+bsqrt(y))` the rationalising factor is a `asqrt(x)-bsqrt(y)` .
If `x=(sqrt(5)+sqrt(3))/(sqrt(80)+sqrt(48)-sqrt(45)-sqrt(27))` then value of `4x^(2)+3x+5` is

Options

A. `15`
B. `2`
C. `12`
D. `5`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Simplify the expression for x
We are given:
\[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{80} + \sqrt{48} - \sqrt{45} - \sqrt{27}} \]

First, we simplify the denominator:
- Calculate each square root:
- \( \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \)
- \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \)
- \( \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \)
- \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \)

Now substitute these values back into the denominator:
\[ \sqrt{80} + \sqrt{48} - \sqrt{45} - \sqrt{27} = 4\sqrt{5} + 4\sqrt{3} - 3\sqrt{5} - 3\sqrt{3} \]

### Step 2: Combine like terms in the denominator
Now, combine the like terms:
- For \( \sqrt{5} \): \( 4\sqrt{5} - 3\sqrt{5} = 1\sqrt{5} = \sqrt{5} \)
- For \( \sqrt{3} \): \( 4\sqrt{3} - 3\sqrt{3} = 1\sqrt{3} = \sqrt{3} \)

Thus, the denominator simplifies to:
\[ \sqrt{5} + \sqrt{3} \]

### Step 3: Substitute back to find x
Now substitute the simplified denominator back into the expression for \( x \):
\[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} \]

Since the numerator and denominator are the same, we have:
\[ x = 1 \]

### Step 4: Substitute x into the equation \( 4x^2 + 3x + 5 \)
Now we substitute \( x = 1 \) into the expression \( 4x^2 + 3x + 5 \):
\[ 4(1)^2 + 3(1) + 5 \]

### Step 5: Calculate the value
Calculating this gives:
\[ 4(1) + 3(1) + 5 = 4 + 3 + 5 = 12 \]

### Final Answer
Thus, the value of \( 4x^2 + 3x + 5 \) is:
\[ \boxed{12} \]