Multiplying Polynomials Maze worksheet for algebra practice.

To solve the problem, we need to multiply the polynomials in each square and use the results to navigate through the maze. Let's start from the "START HERE!" square and work step by step.

Step 1: Start Here!

The starting polynomial is:
\[ 4x^3(2x^5 - 10) \]

#### Multiply:
\[ 4x^3 \cdot 2x^5 = 8x^8 \]
\[ 4x^3 \cdot (-10) = -40x^3 \]

So, the result is:
\[ 8x^8 - 40x^3 \]

This matches the arrow pointing to the next square:
\[ (x - 5)(x + 8) \]

Step 2: Next Square

The next polynomial is:
\[ (x - 5)(x + 8) \]

#### Multiply using the distributive property (FOIL):
\[ x \cdot x = x^2 \]
\[ x \cdot 8 = 8x \]
\[ -5 \cdot x = -5x \]
\[ -5 \cdot 8 = -40 \]

Combine like terms:
\[ x^2 + 8x - 5x - 40 = x^2 + 3x - 40 \]

This matches the arrow pointing to the next square:
\[ 5xy(x + 4y) \]

Step 3: Next Square

The next polynomial is:
\[ 5xy(x + 4y) \]

#### Multiply:
\[ 5xy \cdot x = 5x^2y \]
\[ 5xy \cdot 4y = 20xy^2 \]

So, the result is:
\[ 5x^2y + 20xy^2 \]

This matches the arrow pointing to the next square:
\[ (9x)(2x - 1) \]

Step 4: Next Square

The next polynomial is:
\[ (9x)(2x - 1) \]

#### Multiply:
\[ 9x \cdot 2x = 18x^2 \]
\[ 9x \cdot (-1) = -9x \]

So, the result is:
\[ 18x^2 - 9x \]

This matches the "FINISHED!" square.

Final Answer:

The path through the maze is:
\[ 4x^3(2x^5 - 10) \rightarrow (x - 5)(x + 8) \rightarrow 5xy(x + 4y) \rightarrow (9x)(2x - 1) \]

The final answer is:
\[
\boxed{18x^2 - 9x}
\]