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The question on your screen asks you to divide the polynomial \( p(x) = x^5 - 3x^3 + x^2 - 3x - 5 \) by \( d(x) = x^3 + x^2 + 2 \) using synthetic division to express it in the form \( p(x) = d(x)q(x) + r(x) \), where \( q(x) \) is the quotient and \( r(x) \) is the remainder.

Let’s break it down using synthetic or polynomial long division:

### Step-by-Step Division

We are dividing:

\[
p(x) = x^5 - 3x^3 + x^2 - 3x - 5
\]
by 
\[
d(x) = x^3 + x^2 + 2
\]

1. **First Term of Quotient:** 
   Divide \( x^5 \) by \( x^3 \), which gives \( x^2 \). This is the first term of the quotient, \( q(x) \).

2. **Multiply \( x^2 \) by \( d(x) \):**
   \[
   x^2(x^3 + x^2 + 2) = x^5 + x^4 + 2x^2
   \]
   Now subtract this result from \( p(x) \):

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

3. **Second Term of Quotient:** 
   Divide \( -x^4 \) by \( x^3 \), which gives \( -x \). Add this to the quotient, so \( q(x) = x^2 - x \).

4. **Multiply \( -x \) by \( d(x) \):**
   \[
   -x(x^3 + x^2 + 2) = -x^4 - x^3 - 2x
   \]
   Subtract this result:

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

5. **Third Term of Quotient:** 
   Divide \( -2x^3 \) by \( x^3 \), which gives \( -2 \). So, the quotient \( q(x) = x^2 - x - 2 \).

6. **Multiply \( -2 \) by \( d(x) \):**
   \[
   -2(x^3 + x^2 + 2) = -2x^3 - 2x^2 - 4
   \]
   Subtract this result:

   \[
   (-2x^3 + x^2 - x - 5) - (-2x^3 - 2x^2 - 4) = 3x^2 - x - 1
   \]

### Conclusion:

The quotient is:
\[
q(x) = x^2 - x - 2
\]

The remainder is:
\[
r(x) = 3x^2 - x - 1
\]

Thus, the division gives:

\[
p(x) = (x^3 + x^2 + 2)(x^2 - x - 2) + (3x^2 - x - 1)
\]

Would you like further details or explanations on any specific part?

Here are some related questions you might want to explore:
1. How is synthetic division different from polynomial long division?
2. What are the common mistakes to avoid when dividing polynomials?
3. Can synthetic division be applied to divisors of any degree?
4. How do you interpret the remainder in polynomial division?
5. How can you verify the result of a polynomial division?

**Tip:** Always double-check your signs during division, as a small error can affect the entire quotient and remainder!

Write the polynomial in the form p(x) = d(x)q(x) + r(x). You may use synthetic division wherever applicable. p(x) = x^5 - 3x^3 + x^2 - 3x - 5 ; d(x) = x^3 + x^2 + 2

Learn how to divide the polynomial p(x) = x^5 - 3x^3 + x^2 - 3x - 5 by d(x) = x^3 + x^2 + 2 using synthetic division and express it in the form p(x) = d(x)q(x) + r(x). This problem explores polynomial division step-by-step.

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