Sure, let's factor the polynomial [tex]\( 5x^4 + 20x^3 - 105x^2 \)[/tex] completely, step by step.
1. Identify common factors:
Look for any common factor in all terms of the polynomial. The terms are [tex]\( 5x^4 \)[/tex], [tex]\( 20x^3 \)[/tex], and [tex]\( -105x^2 \)[/tex].
All terms have a common factor of [tex]\( 5x^2 \)[/tex].
So, factor out [tex]\( 5x^2 \)[/tex]:
[tex]\[
5x^2 (x^2 + 4x - 21)
\][/tex]
2. Factor the quadratic expression:
Now, we focus on factoring the quadratic expression [tex]\( x^2 + 4x - 21 \)[/tex].
To factor the quadratic expression, we need to find two numbers that multiply to the constant term (-21) and add to the coefficient of the middle term (4).
- The constant term is -21.
- The middle term's coefficient is 4.
The two numbers that fit these criteria are 7 and -3:
[tex]\[
7 \times (-3) = -21
\][/tex]
[tex]\[
7 + (-3) = 4
\][/tex]
So, the quadratic expression [tex]\( x^2 + 4x - 21 \)[/tex] can be factored as:
[tex]\[
(x + 7)(x - 3)
\][/tex]
3. Write the complete factorization:
Substituting this back into our original factorization:
[tex]\[
5x^2 (x + 7)(x - 3)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( 5x^4 + 20x^3 - 105x^2 \)[/tex] is:
[tex]\[
5x^2(x - 3)(x + 7)
\][/tex]
The correct answer is:
[tex]\[
\boxed{\text{B. } 5x^2(x-3)(x+7)}
\][/tex]
