To find the product of the polynomials [tex]\((-3x^5 - 4x^4)\)[/tex] and [tex]\((7x^2 - 2x + 6)\)[/tex], let's perform a step-by-step polynomial multiplication process.
Start by distributing each term in the first polynomial [tex]\((-3x^5 - 4x^4)\)[/tex] across every term in the second polynomial [tex]\((7x^2 - 2x + 6)\)[/tex].
1. Multiply [tex]\(-3x^5\)[/tex] by each term in the second polynomial:
[tex]\[
-3x^5 \cdot 7x^2 = -21x^7
\][/tex]
[tex]\[
-3x^5 \cdot (-2x) = 6x^6
\][/tex]
[tex]\[
-3x^5 \cdot 6 = -18x^5
\][/tex]
2. Next, multiply [tex]\(-4x^4\)[/tex] by each term in the second polynomial:
[tex]\[
-4x^4 \cdot 7x^2 = -28x^6
\][/tex]
[tex]\[
-4x^4 \cdot (-2x) = 8x^5
\][/tex]
[tex]\[
-4x^4 \cdot 6 = -24x^4
\][/tex]
Now, combine all these results together:
[tex]\[
-21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4
\][/tex]
Next, combine like terms:
1. Combine [tex]\(x^6\)[/tex] terms:
[tex]\[
6x^6 - 28x^6 = -22x^6
\][/tex]
2. Combine [tex]\(x^5\)[/tex] terms:
[tex]\[
-18x^5 + 8x^5 = -10x^5
\][/tex]
As a result, the product of the polynomials is:
[tex]\[
-21x^7 - 22x^6 - 10x^5 - 24x^4
\][/tex]
Therefore, the solution is:
[tex]\[
(-3x^5 - 4x^4)(7x^2 - 2x + 6) = -21x^7 - 22x^6 - 10x^5 - 24x^4
\][/tex]
