https://file.aiquickdraw.com/mathbot/file/17260708582747bsyqa07.jpg

The image shows the expression:

\[
x^3 + 8x^2 + 17x + 10
\]

The problem asks to factor it completely, starting by dividing it by \((x+1)\).

### Steps:
1. **Divide the polynomial \( x^3 + 8x^2 + 17x + 10 \) by \( x + 1 \) using synthetic or polynomial division.**

Let's perform the division:

#### Polynomial Division:

\[
\frac{x^3 + 8x^2 + 17x + 10}{x + 1}
\]

**Step 1:** Divide the first term of the dividend \( x^3 \) by the first term of the divisor \( x \). This gives \( x^2 \).

Multiply \( x^2 \) by \( x + 1 \):
\[
x^2 \cdot (x + 1) = x^3 + x^2
\]

Subtract \( (x^3 + x^2) \) from \( (x^3 + 8x^2 + 17x + 10) \):
\[
(x^3 + 8x^2 + 17x + 10) - (x^3 + x^2) = 7x^2 + 17x + 10
\]

**Step 2:** Divide the first term of the new polynomial \( 7x^2 \) by \( x \), which gives \( 7x \).

Multiply \( 7x \) by \( x + 1 \):
\[
7x \cdot (x + 1) = 7x^2 + 7x
\]

Subtract \( (7x^2 + 7x) \) from \( (7x^2 + 17x + 10) \):
\[
(7x^2 + 17x + 10) - (7x^2 + 7x) = 10x + 10
\]

**Step 3:** Divide the first term \( 10x \) by \( x \), which gives \( 10 \).

Multiply \( 10 \) by \( x + 1 \):
\[
10 \cdot (x + 1) = 10x + 10
\]

Subtract \( (10x + 10) \) from \( (10x + 10) \):
\[
(10x + 10) - (10x + 10) = 0
\]

So, the quotient is:
\[
x^2 + 7x + 10
\]

2. **Now, factor the quotient \( x^2 + 7x + 10 \):**

We look for two numbers that multiply to \( 10 \) and add to \( 7 \). These numbers are \( 5 \) and \( 2 \), so:

\[
x^2 + 7x + 10 = (x + 5)(x + 2)
\]

3. **Final factorization:**

Since \( x^3 + 8x^2 + 17x + 10 = (x + 1)(x^2 + 7x + 10) \), and we've factored \( x^2 + 7x + 10 \), the complete factorization is:

\[
x^3 + 8x^2 + 17x + 10 = (x + 1)(x + 5)(x + 2)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you perform synthetic division instead of polynomial division?
2. How do you factor a cubic polynomial in general?
3. What are some techniques to factor higher degree polynomials?
4. How can the Rational Root Theorem help in factoring polynomials?
5. What is the significance of the remainder in polynomial division?

**Tip:** Always check your factorization by multiplying the factors to ensure you get the original polynomial.

Factor x^3 + 8x^2 + 17x + 10 completely by first dividing by (x+1).

Learn how to factor the polynomial x^3 + 8x^2 + 17x + 10 by first dividing by (x+1). The solution involves polynomial division and factoring techniques. Suitable for high school students in Grades 9-12.

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