We are given the quadratic equation $3x^2 - 7x + 4 = 0$ and asked to find a new quadratic equation whose roots are 2 less than the roots of this original equation.

Step-by-step solution:

1. Find the roots of the original quadratic equation.

   The roots of the quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 3$, $b = -7$, and $c = 4$. Substituting these values into the quadratic formula: $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(3)(4)}}{2(3)}$ Simplifying: $x = \frac{7 \pm \sqrt{49 - 48}}{6} = \frac{7 \pm \sqrt{1}}{6}$ Therefore, the roots are: $x = \frac{7 \pm 1}{6}$ So the two roots are: $x_1 = \frac{7 + 1}{6} = \frac{8}{6} = \frac{4}{3}, \quad x_2 = \frac{7 - 1}{6} = \frac{6}{6} = 1$

2. Find the new roots (subtract 2 from each).

   The new roots are 2 less than the original roots, so: $\text{New root 1} = \frac{4}{3} - 2 = \frac{4}{3} - \frac{6}{3} = \frac{-2}{3}$ $\text{New root 2} = 1 - 2 = -1$

3. Form the new quadratic equation from these new roots.

   If the roots of a quadratic equation are $p$ and $q$, the quadratic can be written as: $a(x - p)(x - q) = 0$ Substituting the new roots $p = \frac{-2}{3}$ and $q = -1$: $3\left(x - \frac{-2}{3}\right)(x - (-1)) = 0$ Simplifying: $3\left(x + \frac{2}{3}\right)(x + 1) = 0$ Multiply out: $3\left[\left(x + \frac{2}{3}\right)(x + 1)\right] = 3\left(x^2 + x + \frac{2}{3}x + \frac{2}{3}\right)$ $= 3\left(x^2 + \frac{5}{3}x + \frac{2}{3}\right)$ $= 3x^2 + 5x + 2 = 0$

The quadratic equation with roots 2 less than the original is:

$3x^2 + 5x + 2 = 0$

Checking the options, this corresponds to none of the options. It seems the available options might have an error, as the derived quadratic equation doesn't match any of the given choices. Would you like a detailed check of the reasoning or to explore similar types of problems?

Here are 5 related questions to consider:

1. How do the signs of the coefficients affect the graph of a quadratic?
2. How can you factor quadratic equations when the roots are fractions?
3. What methods exist to derive quadratic equations from given roots?
4. How does the discriminant of a quadratic indicate the nature of its roots?
5. How does shifting the roots by a constant affect the quadratic form?

Tip: Always cross-check the discriminant when manipulating the roots of a quadratic equation to ensure consistency.