Answer: $391$

Explanation:

$1$ . Rearrange the equation into standard quadratic form ($ax^2 + bx + c = 0$):

   - Distribute the x on the left side:

     $kx^2 - 112x = -8$

   - Add $8$ to both sides to set the equation to zero:

     $kx^2 - 112x + 8 = 0$

$2$ . Understand the condition for distinct real solutions:

   - A quadratic equation has distinct (two different) real solutions if and only if its discriminant ( $b^2 - 4ac$ ) is greater than $0$.

$3$ . Identify the coefficients a, b, and c:

   - From $kx^2 - 112x + 8 = 0$ :

     - $a = k$

     - $b = -112$

     - $c = 8$

$4$ . Set up and solve the discriminant inequality:

   - $b^2 - 4ac > 0$

   - $(-112)^2 - 4(k)(8) > 0$

   - $12544 - 32k > 0$

   - Add $32k$ to both sides:

     $12544 > 32k$

   - Divide by $32$ :

     $\dfrac{12544}{32} > k$

     $392 > k$

$5$ . Find the greatest possible integer value for k:

   - The inequality $k < 392$ means that k must be an integer strictly less than $392$.

   - The greatest integer that satisfies this condition is $391$ .

$6$ . Conclusion:

   - The greatest possible integer value of k is $391$ .