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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.

The quadratic whose roots are two less than the roots of the quadratic equation 3x^2 - 7x + 4 = 0 is…

Learn how to form a new quadratic equation whose roots are two less than the roots of the equation 3x^2 - 7x + 4 = 0. This problem involves using the quadratic formula and root transformation techniques, making it ideal for students in Grades 9-12.

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