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Let's break down the given equation step-by-step:

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

### Step 1: Expand the left-hand side of the equation

We need to expand the product \((x^2 + 1)(2x^3 - 1)\).

Distribute \(x^2 + 1\) over \(2x^3 - 1\):

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

Expand both terms:

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

### Step 2: Substitute back into the original equation

Now substitute the expanded expression into the original equation:

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

### Step 3: Simplify and solve for \(r\)

Now, subtract the terms on the left-hand side from the right-hand side:

\[
2x^5 - x^2 + 2x^3 - 1 + r = 2x^5 + 2x^3 + x^2 + 1
\]

Group like terms on both sides:

\[
(2x^5 - 2x^5) + (2x^3 - 2x^3) + (-x^2 - x^2) + (-1 - 1) + r = 0
\]

This simplifies to:

\[
0 + 0 - 2x^2 - 2 + r = 0
\]

Thus, the equation becomes:

\[
r = 2x^2 + 2
\]

### Final Answer:
The value of \(r\) is:

\[
r = 2x^2 + 2
\]

Would you like further details on any step, or have any questions?

Here are some related questions to explore:
1. What is the general method for expanding binomials?
2. How do we solve equations involving polynomials of higher degrees?
3. How can you check the correctness of a solution like this?
4. What techniques can be used to factor complex polynomial expressions?
5. How does the degree of the polynomial affect its behavior for large values of \(x\)?

**Tip:** When expanding binomials, it’s crucial to distribute each term carefully to avoid errors.

(x^2 + 1)(2x^3 - 1) + r = 2x^5 + 2x^3 + x^2 + 1

Learn how to expand the polynomial expression (x^2 + 1)(2x^3 - 1) and solve for r in the equation. This step-by-step solution covers the distributive property and balancing polynomial equations. Suitable for students in Grades 10-12.

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