Find each product. $4x^2(3x^3+2x^2-5x+1)$

Verified step by step guidance

1. Identify the expression to be multiplied: \$4x^2(3x^3 + 2x^2 - 5x + 1)$.
2. Apply the distributive property by multiplying \$4x^2$ with each term inside the parentheses separately.
3. Multiply \$4x^2\( by the first term: \)3x^3\(. Use the rule for multiplying powers with the same base: \)x^a \cdot x^b = x^{a+b}\(. So, \)4x^2 \cdot 3x^3 = 4 \cdot 3 \cdot x^{2+3} = 12x^5$.
4. Multiply \$4x^2\( by the second term: \)2x^2\(. Similarly, \)4x^2 \cdot 2x^2 = 8x^{2+2} = 8x^4$.
5. Continue by multiplying \$4x^2\( by the third term \)-5x\( and the fourth term \)1$, then write the full expanded expression by combining all these products.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of the other. This process requires applying the distributive property to combine like terms and simplify the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a parenthesis, which is essential when multiplying a monomial by a polynomial.

Combining Like Terms

After multiplying, terms with the same variable raised to the same power are combined by adding or subtracting their coefficients. This step simplifies the polynomial to its standard form.