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Show me solution and steps and the answer

Let's analyze the two equations from the image and break them down to solve.

### Given Equations:
1. \( 6x^5 - 3x^4 + 2x^2 + x + 4 \)
2. \( (3x + 4) \times (6x^5 - 3x^4 + 2x^2 + x + 4) \)

We need to expand and simplify the second equation by distributing \( (3x + 4) \) across the polynomial.

### Step 1: Distribution
We will multiply each term of \( 3x + 4 \) with every term of the second polynomial \( 6x^5 - 3x^4 + 2x^2 + x + 4 \).

#### Multiply \( 3x \) with the polynomial:
\[
3x \times (6x^5 - 3x^4 + 2x^2 + x + 4) = 18x^6 - 9x^5 + 6x^3 + 3x^2 + 12x
\]

#### Multiply \( 4 \) with the polynomial:
\[
4 \times (6x^5 - 3x^4 + 2x^2 + x + 4) = 24x^5 - 12x^4 + 8x^2 + 4x + 16
\]

### Step 2: Combine like terms
Now, we will add the two expanded expressions from above:
\[
(18x^6 - 9x^5 + 6x^3 + 3x^2 + 12x) + (24x^5 - 12x^4 + 8x^2 + 4x + 16)
\]

Combining like terms:
- \( 18x^6 \) remains as it is (no like terms for \( x^6 \)).
- For \( x^5 \), combine \( -9x^5 + 24x^5 = 15x^5 \).
- For \( x^4 \), only \( -12x^4 \) exists.
- For \( x^3 \), we have \( 6x^3 \).
- For \( x^2 \), combine \( 3x^2 + 8x^2 = 11x^2 \).
- For \( x \), combine \( 12x + 4x = 16x \).
- Finally, the constant is \( 16 \).

### Final Expanded Expression:
\[
18x^6 + 15x^5 - 12x^4 + 6x^3 + 11x^2 + 16x + 16
\]

### Answer:
The expanded form of the equation is:
\[
18x^6 + 15x^5 - 12x^4 + 6x^3 + 11x^2 + 16x + 16
\]

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

Here are 5 related questions:
1. How can we factor a polynomial like the one we worked with?
2. What is the degree of the resulting polynomial, and why?
3. How would we solve this equation if it were set equal to zero?
4. Can we use synthetic division to divide this polynomial by a smaller factor?
5. How does the distributive property apply to polynomials?

**Tip:** When multiplying polynomials, always make sure to distribute each term carefully to avoid missing out on any term in the expanded form.

Learn how to expand and simplify the polynomial expression (3x + 4)(6x^5 - 3x^4 + 2x^2 + x + 4). This problem covers key algebraic concepts, including the distributive property and combining like terms. Suitable for students in Grades 9-12.

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