Question

Use long division to divide.

15x^2+14x-8/3x+4

Answer 1

To divide \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, follow these steps:

1. **Set up the long division:** Write \( 15x^2 + 14x - 8 \) under the long division bar and \( 3x + 4 \) outside.

2. **Divide the leading terms:** Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):
\[
\frac{15x^2}{3x} = 5x
\]
Write \( 5x \) above the division bar.

3. **Multiply and subtract:** Multiply \( 5x \) by \( 3x + 4 \) and write it under the dividend:
\[
5x \cdot (3x + 4) = 15x^2 + 20x
\]
Now subtract this from \( 15x^2 + 14x - 8 \):
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]

4. **Bring down the next term:** Since there are no more terms to bring down, we proceed with what we have: \(-6x - 8\).

5. **Repeat the process:** Now divide the leading term \(-6x\) by the leading term \(3x\):
\[
\frac{-6x}{3x} = -2
\]
Write \(-2\) above the division bar.

6. **Multiply and subtract again:** Multiply \(-2\) by \(3x + 4\):
\[
-2 \cdot (3x + 4) = -6x - 8
\]
Now subtract:
\[
(-6x - 8) - (-6x - 8) = 0
\]

At this point, there are no more terms left to bring down, and the remainder is \(0\).

The final result of the long division is:
\[
15x^2 + 14x - 8 \div (3x + 4) = 5x - 2
\]

Thus, the quotient is \( \boxed{5x - 2} \).