To determine the completely factored form of the polynomial [tex]\(5x^4 + 20x^3 - 105x^2\)[/tex], let's go through the process step-by-step.
1. First, factor out the greatest common factor (GCF) of the polynomial terms.
All terms have [tex]\(x^2\)[/tex] and a common factor of 5:
[tex]\[
5x^4 + 20x^3 - 105x^2 = 5x^2(x^2 + 4x - 21)
\][/tex]
2. Next, we need to factor the quadratic polynomial [tex]\(x^2 + 4x - 21\)[/tex].
We look for two numbers that multiply to [tex]\(-21\)[/tex] (the constant term) and add up to [tex]\(4\)[/tex] (the coefficient of [tex]\(x\)[/tex]):
These numbers are [tex]\(7\)[/tex] and [tex]\(-3\)[/tex], since:
[tex]\[
7 \times (-3) = -21 \quad \text{and} \quad 7 + (-3) = 4
\][/tex]
So, we can factor [tex]\(x^2 + 4x - 21\)[/tex] as:
[tex]\[
x^2 + 4x - 21 = (x - 3)(x + 7)
\][/tex]
3. Putting it all together, we factor the original polynomial completely:
[tex]\[
5x^4 + 20x^3 - 105x^2 = 5x^2(x^2 + 4x - 21) = 5x^2(x - 3)(x + 7)
\][/tex]
Therefore, the completely factored form of the polynomial is:
[tex]\[
5x^2(x - 3)(x + 7)
\][/tex]
This matches with option B.
Hence, the correct answer is:
B. [tex]\(5x^2(x-3)(x+7)\)[/tex]
