Answer :
We start with the polynomial
[tex]$$-5x^4 + 70x^3 + 75x^2.$$[/tex]
Step 1. Factor out the greatest common factor
Observe that each term has a factor of [tex]$5x^2$[/tex]. In fact, we can choose to factor out [tex]$-5x^2$[/tex] for a cleaner quadratic part. Factoring out [tex]$-5x^2$[/tex], we have
[tex]$$
-5x^4 + 70x^3 + 75x^2 = -5x^2 \left( \frac{-5x^4 + 70x^3 + 75x^2}{-5x^2} \right).
$$[/tex]
Dividing each term by [tex]$-5x^2$[/tex] yields
[tex]$$
\frac{-5x^4}{-5x^2} = x^2, \quad \frac{70x^3}{-5x^2} = -14x, \quad \frac{75x^2}{-5x^2} = -15.
$$[/tex]
Thus, the expression inside the parentheses becomes
[tex]$$
x^2 - 14x - 15.
$$[/tex]
So the polynomial can be written as
[tex]$$
-5x^2 \left( x^2 - 14x - 15 \right).
$$[/tex]
Step 2. Factor the quadratic [tex]$x^2 - 14x - 15$[/tex]
We look for two numbers that multiply to [tex]$-15$[/tex] (the constant term) and add to [tex]$-14$[/tex] (the coefficient of [tex]$x$[/tex]). Notice that
[tex]$$
(-15) \cdot 1 = -15 \quad \text{and} \quad (-15) + 1 = -14.
$$[/tex]
Thus, the quadratic factors as
[tex]$$
x^2 - 14x - 15 = (x - 15)(x + 1).
$$[/tex]
Step 3. Write the final factored form
Substituting the factorization of the quadratic back, we get
[tex]$$
-5x^4 + 70x^3 + 75x^2 = -5x^2\,(x-15)(x+1).
$$[/tex]
This is the completely factored form of the polynomial.
[tex]$$-5x^4 + 70x^3 + 75x^2.$$[/tex]
Step 1. Factor out the greatest common factor
Observe that each term has a factor of [tex]$5x^2$[/tex]. In fact, we can choose to factor out [tex]$-5x^2$[/tex] for a cleaner quadratic part. Factoring out [tex]$-5x^2$[/tex], we have
[tex]$$
-5x^4 + 70x^3 + 75x^2 = -5x^2 \left( \frac{-5x^4 + 70x^3 + 75x^2}{-5x^2} \right).
$$[/tex]
Dividing each term by [tex]$-5x^2$[/tex] yields
[tex]$$
\frac{-5x^4}{-5x^2} = x^2, \quad \frac{70x^3}{-5x^2} = -14x, \quad \frac{75x^2}{-5x^2} = -15.
$$[/tex]
Thus, the expression inside the parentheses becomes
[tex]$$
x^2 - 14x - 15.
$$[/tex]
So the polynomial can be written as
[tex]$$
-5x^2 \left( x^2 - 14x - 15 \right).
$$[/tex]
Step 2. Factor the quadratic [tex]$x^2 - 14x - 15$[/tex]
We look for two numbers that multiply to [tex]$-15$[/tex] (the constant term) and add to [tex]$-14$[/tex] (the coefficient of [tex]$x$[/tex]). Notice that
[tex]$$
(-15) \cdot 1 = -15 \quad \text{and} \quad (-15) + 1 = -14.
$$[/tex]
Thus, the quadratic factors as
[tex]$$
x^2 - 14x - 15 = (x - 15)(x + 1).
$$[/tex]
Step 3. Write the final factored form
Substituting the factorization of the quadratic back, we get
[tex]$$
-5x^4 + 70x^3 + 75x^2 = -5x^2\,(x-15)(x+1).
$$[/tex]
This is the completely factored form of the polynomial.
