Use the long division method to find the result when 4, x, cubed, plus, 12, x, squared, plus, 19, x, plus, 74x

Question

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:

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