To find the product of the polynomials [tex]\((4x - 3)(2x^2 - 7x + 1)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials). Let's perform the multiplication step-by-step.
Given:
[tex]\[ (4x - 3)(2x^2 - 7x + 1) \][/tex]
First, distribute [tex]\(4x\)[/tex] to each term in the second polynomial:
[tex]\[ 4x \cdot 2x^2 = 8x^3 \][/tex]
[tex]\[ 4x \cdot (-7x) = -28x^2 \][/tex]
[tex]\[ 4x \cdot 1 = 4x \][/tex]
Next, distribute [tex]\(-3\)[/tex] to each term in the second polynomial:
[tex]\[ -3 \cdot 2x^2 = -6x^2 \][/tex]
[tex]\[ -3 \cdot (-7x) = 21x \][/tex]
[tex]\[ -3 \cdot 1 = -3 \][/tex]
Now, combine all these terms:
[tex]\[ 8x^3 - 28x^2 + 4x - 6x^2 + 21x - 3 \][/tex]
Group together the like terms:
[tex]\[ 8x^3 + (-28x^2 - 6x^2) + (4x + 21x) - 3 \][/tex]
[tex]\[ 8x^3 - 34x^2 + 25x - 3 \][/tex]
So, the product of [tex]\((4x - 3)(2x^2 - 7x + 1)\)[/tex] is:
[tex]\[ 8x^3 - 34x^2 + 25x - 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{8x^3 - 34x^2 + 25x - 3} \][/tex]
