How many integers satisfy the inequality x(3x-4) is less than or equal to 6x^2-3x+5/10

Why write 5/10 instead of 1/2 or 0.5?

Perhaps you mean (6x^2- 3x+ 5)/10? Saying that $x(3x- 4)\le (6x^2- 3x+ 5)/10$ is the same as saying $10x(3x- 4)\le 6x^2- 3x+ 5$ which in turn is the same as $30x^2- 40x- (6x^2- 3x+ 5)= 24x^2- 37x- 5\le 0$. You can use the quadratic formula to determine where that polynoial is equal to 0. It will be less than 0 between those two values so it is only necessary to determine how many integers there are between the two roots of $24x^2- 37x- 5$.

You might note that $24x^2- 37x- 5=(8x+1)(3x-5)$
You might note that $24x^2- 37x- 5=(8x+1)(3x-5)$