The polynomial 3x² + 10x + 8 has a factor of 3x + 4. What is the other factor of 3x² + 10x + 8?

To find the other factor of the polynomial 3x² + 10x + 8, we start from the information given that one of its factors is 3x + 4.

We can use polynomial long division or synthetic division to divide 3x² + 10x + 8 by 3x + 4.

Here’s how polynomial long division works in this case:

1. Divide the leading term of the dividend (3x²) by the leading term of the divisor (3x), which gives us x.
2. Multiply the entire divisor (3x + 4) by this result (x): 3x * x + 4 * x = 3x² + 4x.
3. Subtract this result from the original polynomial:
• (3x² + 10x + 8) – (3x² + 4x) = 6x + 8
4. Next, divide the leading term of the result (6x) by the leading term of the divisor (3x), giving us (2).
5. Multiply the divisor by (2): (3x + 4) * 2 = 6x + 8.
6. Subtract this again from the result: (6x + 8) – (6x + 8) = 0.

This indicates that we had no remainder, and thus, the division is complete.

The result from our division is x + 2.

Thus, the other factor of the polynomial 3x² + 10x + 8 is:

x + 2.

In summary, we have factored the polynomial as:

(3x + 4)(x + 2).