Divide.
StartFraction 5 x cubed minus 2 x squared plus 2 x plus 3 Over x squared plus x plus 1 EndFraction
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Part 1
StartFraction 5 x cubed minus 2 x squared plus 2 x plus 3 Over x squared plus x plus 1 EndFraction
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To divide the expression

\[
\frac{5x^3 - 2x^2 + 2x + 3}{x^2 + x + 1}
\]

we perform polynomial long division. Here’s the step-by-step process:

### Step 1: Divide the first term
Divide the leading term of the numerator \(5x^3\) by the leading term of the denominator \(x^2\):

\[
\frac{5x^3}{x^2} = 5x
\]

This is the first term of the quotient.

### Step 2: Multiply and subtract
Now multiply \(5x\) by the entire denominator \(x^2 + x + 1\):

\[
5x \cdot (x^2 + x + 1) = 5x^3 + 5x^2 + 5x
\]

Now subtract this from the original polynomial:

\[
(5x^3 - 2x^2 + 2x + 3) - (5x^3 + 5x^2 + 5x) = -7x^2 - 3x + 3
\]

### Step 3: Divide the new leading term
Divide the new leading term \(-7x^2\) by \(x^2\):

\[
\frac{-7x^2}{x^2} = -7
\]

This is the next term of the quotient.

### Step 4: Multiply and subtract again
Now multiply \(-7\) by the entire denominator \(x^2 + x + 1\):

\[
-7 \cdot (x^2 + x + 1) = -7x^2 - 7x - 7
\]

Subtract this from the result of the previous step:

\[
(-7x^2 - 3x + 3) - (-7x^2 - 7x - 7) = 4x + 10
\]

### Final result
Now, the quotient is \(5x - 7\) with a remainder of \(4x + 10\). So, we can write the result as:

\[
\frac{5x^3 - 2x^2 + 2x + 3}{x^2 + x + 1} = 5x - 7 + \frac{4x + 10}{x^2 + x + 1}
\]

Would you like more details on any of the steps or further clarification?

### Related questions:
1. How does polynomial long division differ from synthetic division?
2. Can this process be applied to polynomials of any degree?
3. What are some examples of real-world applications for polynomial division?
4. How can you check if the remainder is correct in polynomial division?
5. What happens if the remainder is 0 in polynomial division?

**Tip:** Always align the terms by their degrees during polynomial long division to avoid mistakes.

Divide. StartFraction 5x cubed minus 2x squared plus 2x plus 3 Over x squared plus x plus 1 EndFraction

This guide explains how to divide the polynomial 5x^3 - 2x^2 + 2x + 3 by x^2 + x + 1 using polynomial long division. Step-by-step solutions break down the key concepts of algebra, including division of cubic and quadratic polynomials. Suitable for Grades 10-12.

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