Question

Use the long division method to find the result when 4, x, cubed, plus, 12, x, squared, plus, 19, x, plus, 74x
3
+12x
2
+19x+7 is divided by 2, x, plus, 12x+1.



Solve the problem just like the example below
⬇️
Use the long division method to find the result when 4, x, cubed, plus, 5, x, squared, plus, 5, x, plus, 14x
3
+5x
2
+5x+1 is divided by 4, x, plus, 14x+1.
Use Long Division:
Use Long Division:
x, squaredx
2
plus, x+x plus, 1+1
4, x4x plus, 1+1 right parenthesis)4, x, cubed4x
3
plus, 5, x, squared+5x
2
plus, 5, x+5x plus, 1+1
start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, x, squared+
−
x
2

4, x, squared4x
2
plus, 5, x+5x plus, 1+1
start superscript, minus, end superscript, 4, x, squared
−
4x
2
plus, start superscript, minus, end superscript, x+
−
x
4, x4x plus, 1+1
start superscript, minus, end superscript, 4, x
−
4x plus, start superscript, minus, end superscript, 1+
−
1
00
Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

Let's perform the long division of \( 4x^3 + 12x^2 + 19x + 7 \) by \( 2x + 12 \).

**Use Long Division:**
\[
\begin{array}{r|l}
& 2x^2 + 6x + 1 \\
\hline
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
& - (4x^3 + 24x^2) \\
\hline
& -12x^2 + 19x + 7 \\
& + (12x^2 + 72x) \\
\hline
& -53x + 7 \\
& + (53x + 636) \\
\hline
& -629 \\
\end{array}
\]

1. **Divide the first term:**
\[
\frac{4x^3}{2x} = 2x^2
\]

2. **Multiply:**
\[
2x^2 \cdot (2x + 12) = 4x^3 + 24x^2
\]

3. **Subtract:**
\[
(4x^3 + 12x^2) - (4x^3 + 24x^2) = -12x^2
\]

4. **Bring down the next term:**
\[
-12x^2 + 19x
\]

5. **Divide the first term:**
\[
\frac{-12x^2}{2x} = -6x
\]

6. **Multiply:**
\[
-6x \cdot (2x + 12) = -12x^2 - 72x
\]

7. **Subtract:**
\[
(-12x^2 + 19x) - (-12x^2 - 72x) = 91x
\]

8. **Bring down the last term:**
\[
91x + 7
\]

9. **Divide the first term:**
\[
\frac{91x}{2x} = 45.5 \text{ (Or } \frac{91}{2}\text{)}
\]

10. **Multiply:**
\[
45.5 \cdot (2x + 12) = 91x + 546
\]

11. **Subtract:**
\[
(91x + 7) - (91x + 546) = -539
\]

So, the final result of the long division is:

\[
\text{Quotient: } 2x^2 - 6x + 45.5
\]
\[
\text{Remainder: } -539
\]

Thus, we can express the answer as:

\[
\text{Answer: } 2x^2 - 6x + 45.5 \quad \text{with remainder } -539
\]