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Answer :
We want to factor the polynomial
[tex]$$
4x^4 - 28x^3 - 120x^2.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Each term in the polynomial has a common factor of [tex]$4x^2$[/tex]. Factoring this out, we have
[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 \left( x^2 - 7x - 30 \right).
$$[/tex]
Step 2. Factor the Quadratic Expression:
We now focus on the quadratic expression
[tex]$$
x^2 - 7x - 30.
$$[/tex]
To factor this, we need to find two numbers whose product is [tex]$-30$[/tex] and whose sum is [tex]$-7$[/tex]. The two numbers that satisfy these conditions are [tex]$-10$[/tex] and [tex]$3$[/tex], since
[tex]$$
-10 \times 3 = -30 \quad \text{and} \quad -10 + 3 = -7.
$$[/tex]
Thus, the quadratic factors as
[tex]$$
x^2 - 7x - 30 = (x - 10)(x + 3).
$$[/tex]
Step 3. Write the Completely Factored Form:
Substitute the factorization of the quadratic back into the expression:
[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 (x - 10)(x + 3).
$$[/tex]
This is the completely factored form of the given polynomial.
[tex]$$
4x^4 - 28x^3 - 120x^2.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Each term in the polynomial has a common factor of [tex]$4x^2$[/tex]. Factoring this out, we have
[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 \left( x^2 - 7x - 30 \right).
$$[/tex]
Step 2. Factor the Quadratic Expression:
We now focus on the quadratic expression
[tex]$$
x^2 - 7x - 30.
$$[/tex]
To factor this, we need to find two numbers whose product is [tex]$-30$[/tex] and whose sum is [tex]$-7$[/tex]. The two numbers that satisfy these conditions are [tex]$-10$[/tex] and [tex]$3$[/tex], since
[tex]$$
-10 \times 3 = -30 \quad \text{and} \quad -10 + 3 = -7.
$$[/tex]
Thus, the quadratic factors as
[tex]$$
x^2 - 7x - 30 = (x - 10)(x + 3).
$$[/tex]
Step 3. Write the Completely Factored Form:
Substitute the factorization of the quadratic back into the expression:
[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 (x - 10)(x + 3).
$$[/tex]
This is the completely factored form of the given polynomial.
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