Multiplying Polynomials
We can use the foil method to multiply polynomials. When using the foil method, we are using the distributive and commutative rules to distribute the terms from one of the polynomials being multiplied to the other.

Sample:

(x^2 - 2x + 1) ( 3x^3 + 4x^2 + 5x)

Note: (x^2 - 2x + 1) is the first expression and (3x^3 + 4x^2 + 5x)
is the second expression. We will multiply every term of the first expression by every term of the second expression and then combine all like terms to simplify.

Step 1: Multiply x^2 by every term in the second expression.

x^2 (3x^3 + 4x^2 + 5x) = 3x^5 + 4x^4 + 5x^3.

Step 2: Multiply -2x by every term in the second expression.

-2x (3x^3 + 4x^2 + 5x) = -6x^4 - 8x^3 - 10x^2.

Step 3: Multiply 1 by every term in the second expression.

1 (3x^3 + 4x^2 + 5x) = 3x^3 + 4x^2 + 5x.

Step 4: Combine all like terms to simplify.

3x^5 + 4x^4 + 5x^3 - 6x^4 - 8x^3 - 10x^2 + 3x^3 + 4x^2 + 5x

Final answer: 3x^5 - 2x^4 - 6x^2 + 5x

NOTE: The terms 5x^3, -8x^3 and 3x^3 were cancelled because their combination will produce zero.