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We are given the quadratic equation:

\[
2x^2 + px + 3 = 0
\]

It is also stated that one of the roots is \( x = 4 \). We are tasked with finding the other root.

### Step 1: Use Vieta's Formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is given by:

\[
\text{Sum of the roots} = -\frac{b}{a}
\]

and the product of the roots is:

\[
\text{Product of the roots} = \frac{c}{a}
\]

For the given equation \( 2x^2 + px + 3 = 0 \):
- \( a = 2 \)
- \( b = p \)
- \( c = 3 \)

Let the two roots be \( x_1 = 4 \) (as given) and \( x_2 \) (the unknown root). Using the formulas:

### Step 2: Use the Product of Roots Formula
The product of the roots is:

\[
x_1 \cdot x_2 = \frac{c}{a} = \frac{3}{2}
\]

Substitute \( x_1 = 4 \):

\[
4 \cdot x_2 = \frac{3}{2}
\]

Solve for \( x_2 \):

\[
x_2 = \frac{3}{2} \div 4 = \frac{3}{2} \cdot \frac{1}{4} = \frac{3}{8}
\]

Thus, the other root is \( \frac{3}{8} \), which corresponds to option B.

### Final Answer:
The other root is \( \frac{3}{8} \), so the correct answer is **B**.

Would you like further details or have any questions?

### Related Questions:
1. How do you derive Vieta’s formulas?
2. What are the applications of Vieta's formulas beyond quadratic equations?
3. How do we solve a quadratic equation when both roots are given?
4. Can the discriminant be used to verify the nature of the roots in this case?
5. How do you factor quadratic equations using the roots?

### Tip:
Vieta’s formulas provide a quick way to check relationships between the coefficients of a polynomial and its roots without solving the equation directly.

2x^2 + px + 3 = 0 has 4 as one of the roots. What is the other root?

Find the other root of the quadratic equation 2x^2 + px + 3 = 0 when one root is 4. The solution uses Vieta's formulas to determine the sum and product of the roots. Suitable for students in Grades 8-10.

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