Find each product. See Example 5. 4x² (3x³ + 2x² - 5x +1)

Verified step by step guidance

1. Identify the expression to be multiplied: \(4x^{2} \times (3x^{3} + 2x^{2} - 5x + 1)\).
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}$. This gives \(4x^{2} \times 3x^{3} = 12x^{5}\).
4. Multiply \$4x^{2}\( by the second term: \)2x^{2}$. This gives \(4x^{2} \times 2x^{2} = 8x^{4}\).
5. Continue by multiplying \$4x^{2}\( by the third term \)-5x$ and the fourth term \(1\), writing each product before combining all terms.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of the other. This process requires multiplying coefficients and adding exponents of like bases, then combining like terms to simplify the expression.

Exponent Rules

When multiplying terms with the same base, add their exponents (e.g., x^a * x^b = x^(a+b)). This rule is essential for correctly handling powers of variables during polynomial multiplication.

Combining Like Terms

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