Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2 + 3a) x^2 + \left( \frac{a + 2}{a - 1} \right) x + b, a \neq 1$. If $f(x + y) = f(x) + f(y) + 1 - \frac{2}{7} xy$, then the value of $28 \sum_{i=1}^{5} |f(i)|$ is

Step 1: Analyze the given functional equation

The problem provides a functional equation relating $f(x+y)$ to $f(x)$ and $f(y)$. This equation is key to determining the coefficients of the quadratic function $f(x)$. We will substitute $x=0$ and $y=0$ into this equation to find the value of $b$. Then, we will substitute the general form of $f(x)$ into the functional equation to find the values of $a$ and $b$.

Step 2: Determine the constant term 'b'

Substitute $x=0$ and $y=0$ into the functional equation $f(x+y) = f(x) + f(y) + 1 - \frac{2}{7} xy$. This simplifies to $f(0) = f(0) + f(0) + 1$, which implies $f(0) = -1$. From the definition of $f(x)$, we know that $f(0) = b$. Therefore, we find that $b = -1$.

Step 3: Substitute $f(x)$ into the functional equation

Now, substitute the expression for $f(x)$ into the functional equation. Expand both sides and simplify. This will allow us to compare the coefficients of $xy$ and other terms to find the value of $a$.

Step 4: Solve for 'a'

Expand the left side of the equation. After simplification, the terms involving $x^2$, $y^2$, $x$, and $y$ cancel out on both sides. We are left with a comparison of the $xy$ terms. Equating the coefficients of $xy$ gives $2(2+3a) = -\frac{2}{7}$. Solving this equation for $a$ yields $a = -\frac{5}{7}$.

Step 5: Determine the final form of $f(x)$

Now that we have the values of $a = -\frac{5}{7}$ and $b = -1$, we can substitute them back into the original definition of $f(x)$. Calculate the coefficients $(2+3a)$ and $\left(\frac{a+2}{a-1}\right)$ to get the complete expression for $f(x)$.

Step 6: Calculate $\sum_{i=1}^{5} |f(i)|$

Now we need to calculate $f(i)$ for $i=1, 2, 3, 4, 5$. Since all the coefficients of $f(x)$ are negative and the constant term is negative, $f(x)$ will be negative for positive values of $x$. Therefore, $|f(i)| = -f(i)$. Calculate each $f(i)$ and sum their absolute values.

Step 7: Calculate the final value

Finally, multiply the sum of the absolute values by 28 to get the required answer.

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