Question

Use the long division method to find the result when x, cubed, plus, 5, x, squared, plus, 8, x, plus, 4x
3
+5x
2
+8x+4 is divided by x, plus, 2x+2.



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, 1
4x
3
+5x
2
+5x+1 is divided by
4, x, plus, 1
4x+1.
Use Long Division:
Use Long Division:


x, squared
x
2
plus, x
+x
plus, 1
+1


4, x
4x
plus, 1
+1
right parenthesis
)
4, x, cubed
4x
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, squared
4x
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, x
4x
plus, 1
+1





start superscript, minus, end superscript, 4, x
−
4x
plus, start superscript, minus, end superscript, 1
+
−
1






0
0

Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

Let's perform long division to divide \(x^3 + 5x^2 + 8x + 4\) by \(x + 2\).

**Step 1: Setup the division**

We are dividing:
\[
\begin{array}{r|l}
x + 2 & x^3 + 5x^2 + 8x + 4 \\
\end{array}
\]

**Step 2: Divide the leading terms**

Divide the leading term \(x^3\) by the leading term \(x\):
\[
\frac{x^3}{x} = x^2
\]

**Step 3: Multiply and subtract**

Now, multiply \(x^2\) by \(x + 2\) and subtract from the original polynomial:
\[
x^2(x + 2) = x^3 + 2x^2
\]

Subtract:
\[
(x^3 + 5x^2 + 8x + 4) - (x^3 + 2x^2) = (5x^2 - 2x^2) + 8x + 4 = 3x^2 + 8x + 4
\]

So far, we have:
\[
\begin{array}{r|l}
x + 2 & x^3 + 5x^2 + 8x + 4 \\
& - (x^3 + 2x^2) \\
\hline
& 3x^2 + 8x + 4 \\
\end{array}
\]

**Step 4: Repeat the process**

Now we continue with \(3x^2 + 8x + 4\).

Divide \(3x^2\) by \(x\):
\[
\frac{3x^2}{x} = 3x
\]

Multiply \(3x\) by \(x + 2\) and subtract:
\[
3x(x + 2) = 3x^2 + 6x
\]

Subtract:
\[
(3x^2 + 8x + 4) - (3x^2 + 6x) = (8x - 6x) + 4 = 2x + 4
\]

So far, we have:
\[
\begin{array}{r|l}
x + 2 & x^3 + 5x^2 + 8x + 4 \\
& - (x^3 + 2x^2) \\
\hline
& 3x^2 + 8x + 4 \\
& - (3x^2 + 6x) \\
\hline
& 2x + 4 \\
\end{array}
\]

**Step 5: Repeat again**

Now we continue with \(2x + 4\).

Divide \(2x\) by \(x\):
\[
\frac{2x}{x} = 2
\]

Multiply \(2\) by \(x + 2\) and subtract:
\[
2(x + 2) = 2x + 4
\]

Subtract:
\[
(2x + 4) - (2x + 4) = 0
\]

So we have:
\[
\begin{array}{r|l}
x + 2 & x^3 + 5x^2 + 8x + 4 \\
& - (x^3 + 2x^2) \\
\hline
& 3x^2 + 8x + 4 \\
& - (3x^2 + 6x) \\
\hline
& 2x + 4 \\
& - (2x + 4) \\
\hline
& 0 \\
\end{array}
\]

**Final Result:**

The division is complete, and the result is:
\[
x^2 + 3x + 2
\]

**Answer:** \(x^2 + 3x + 2\)