When the product of 6 and the square of a number is increased by 5 times the number, the result is 4. Select all of the values that the number could be -(4)/(3) (1)/(2) -(3)/(4)

Let's denote the unknown number as $x$. According to the given information, we can set up the equation:<br /><br />$6x^2 + 5x = 4$<br /><br />To solve this quadratic equation, we can rearrange it to standard form:<br /><br />$6x^2 + 5x - 4 = 0$<br /><br />Now, we can use the quadratic formula to find the values of $x$:<br /><br />$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />In this case, $a = 6$, $b = 5$, and $c = -4$. Plugging these values into the quadratic formula, we get:<br /><br />$x = \frac{-5 \pm \sqrt{5^2 - 4(6)(-4)}}{2(6)}$<br /><br />$x = \frac{-5 \pm \sqrt{25 + 96}}{12}$<br /><br />$x = \frac{-5 \pm \sqrt{121}}{12}$<br /><br />$x = \frac{-5 \pm 11}{12}$<br /><br />So, the two possible values for $x$ are:<br /><br />$x_1 = \frac{-5 + 11}{12} = \frac{6}{12} = \frac{1}{2}$<br /><br />$x_2 = \frac{-5 - 11}{12} = \frac{-16}{12} = -\frac{4}{3}$<br /><br />Therefore, the values that the number could be are $\frac{1}{2}$ and $-\frac{4}{3}$.