To solve the multiplication of the polynomials [tex]\(-3x^2\)[/tex] and [tex]\((6x^2 + 2x - 3)\)[/tex], let's work through it step-by-step:
1. Distribute [tex]\(-3x^2\)[/tex] to each term in the polynomial [tex]\((6x^2 + 2x - 3)\)[/tex].
We will multiply [tex]\(-3x^2\)[/tex] by each of the terms in the polynomial individually:
[tex]\[
-3x^2 \cdot 6x^2
\][/tex]
[tex]\[
-3x^2 \cdot 2x
\][/tex]
[tex]\[
-3x^2 \cdot (-3)
\][/tex]
2. Perform the multiplications.
First, multiply [tex]\(-3x^2\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[
-3x^2 \cdot 6x^2 = -18x^4
\][/tex]
Next, multiply [tex]\(-3x^2\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
-3x^2 \cdot 2x = -6x^3
\][/tex]
Finally, multiply [tex]\(-3x^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-3x^2 \cdot (-3) = 9x^2
\][/tex]
3. Combine all the terms to write the final expanded polynomial:
[tex]\[
-18x^4 - 6x^3 + 9x^2
\][/tex]
Therefore, after multiplying [tex]\(-3x^2\)[/tex] by [tex]\((6x^2 + 2x - 3)\)[/tex], the resulting polynomial is:
[tex]\[
-18x^4 - 6x^3 + 9x^2
\][/tex]
