If `(2x+3y+4) (2x+3y-5)` is equal to `(ax^(2)+by^(2) + 2hxy + 2gx +2fy +c)`. Then what is the value of `{3(g-f-c)//ab}` ?

Options

A. `(41)/(24)`
B. 1
C. `(31)/(24)`
D. `(25)/(24)`

Answer

To solve the problem, we need to expand the expression \((2x + 3y + 4)(2x + 3y - 5)\) and compare it with the expression \(ax^2 + by^2 + 2hxy + 2gx + 2fy + c\).

### Step-by-step Solution:

1. **Expand the Expression**:
We start by expanding \((2x + 3y + 4)(2x + 3y - 5)\).

\[ = (2x + 3y + 4)(2x + 3y) + (2x + 3y + 4)(-5) \]

First, calculate \((2x + 3y)(2x + 3y)\):
\[ = 4x^2 + 6xy + 6xy + 9y^2 = 4x^2 + 12xy + 9y^2 \]

Now calculate \((2x + 3y + 4)(-5)\):
\[ = -10x - 15y - 20 \]

Combine both results:
\[ = 4x^2 + 12xy + 9y^2 - 10x - 15y - 20 \]

So, the expanded expression is:
\[ 4x^2 + 9y^2 + 12xy - 10x - 15y - 20 \]

2. **Match Coefficients**:
Now we compare the coefficients of the expanded expression with \(ax^2 + by^2 + 2hxy + 2gx + 2fy + c\).

- From \(4x^2\): \(a = 4\)
- From \(9y^2\): \(b = 9\)
- From \(12xy\): \(2h = 12 \Rightarrow h = 6\)
- From \(-10x\): \(2g = -10 \Rightarrow g = -5\)
- From \(-15y\): \(2f = -15 \Rightarrow f = -\frac{15}{2}\)
- From \(-20\): \(c = -20\)

3. **Calculate the Required Expression**:
We need to find the value of \(\frac{3(g - f - c)}{ab}\).

Substitute the values we found:
\[ g = -5, \quad f = -\frac{15}{2}, \quad c = -20 \]

Calculate \(g - f - c\):
\[ g - f - c = -5 - \left(-\frac{15}{2}\right) - (-20) \]
\[ = -5 + \frac{15}{2} + 20 \]
Convert \(-5\) and \(20\) to fractions:
\[ = -\frac{10}{2} + \frac{15}{2} + \frac{40}{2} = \frac{-10 + 15 + 40}{2} = \frac{45}{2} \]

Now, calculate \(ab\):
\[ ab = 4 \times 9 = 36 \]

Finally, substitute into the expression:
\[ \frac{3(g - f - c)}{ab} = \frac{3 \times \frac{45}{2}}{36} = \frac{\frac{135}{2}}{36} = \frac{135}{72} = \frac{15}{8} \]

### Final Result:
The value of \(\frac{3(g - f - c)}{ab}\) is \(\frac{15}{8}\).