What is the factored form of 6x^4-21x^3-12x^2 ?

thank you

To find the factored form of the quadratic expression 6x^4 - 21x^3 - 12x^2, we need to factor out the greatest common factor (GCF) from each term. In this case, the GCF is 3x^2. By factoring out 3x^2, we get:

3x^2(2x^2 - 7x - 4)

Now, we need to factor the quadratic term 2x^2 - 7x - 4. To do this, we look for two numbers that multiply to give -8 (the product of the leading coefficient 2 and the constant term -4) and add up to -7 (the coefficient of the middle term). These numbers are -8 and 1.

So, we can rewrite the quadratic term as:

2x^2 - 7x - 4 = 2x^2 - 8x + x - 4

Grouping the terms, we get:

2x(x - 4) + 1(x - 4)

Now, we can factor out the common binomial factor (x - 4), giving us the final factored form:

3x^2(x - 4)(2x + 1)

To find the factored form of the given polynomial, we need to factor out the common terms that appear in each term of the polynomial.

The given polynomial is: 6x^4 - 21x^3 - 12x^2

Step 1: Factor out the greatest common factor.
The greatest common factor of the terms 6x^4, -21x^3, and -12x^2 is 3x^2.
Factor out 3x^2 from each term:

3x^2(2x^2 - 7x - 4)

Step 2: Factor the quadratic expression (2x^2 - 7x - 4).
To factor the quadratic expression, we need to find two binomials whose product is equal to the quadratic expression. These binomials can be found by finding two numbers whose product is equal to the product of the first and last coefficients (2 * -4 = -8) and whose sum is equal to the coefficient of the middle term (-7).

The numbers that satisfy these conditions are -8 and 1, since (-8 * 1 = -8) and (-8 + 1 = -7).

Split the middle term (-7x) using -8x and +x:

3x^2(2x^2 - 8x + x - 4)

Step 3: Group and factor by grouping.
Group the terms and factor by grouping:

3x^2((2x^2 - 8x) + (x - 4))

Factor out the common terms from each group:

3x^2(2x(x - 4) + (x - 4))

Step 4: Factor out the common binomial.
In this step, we notice that we have a common binomial, (x - 4), in each group. We can factor it out.

3x^2(x - 4)(2x + 1)

Therefore, the factored form of the given polynomial 6x^4 - 21x^3 - 12x^2 is 3x^2(x - 4)(2x + 1).

3x^2(2x^2 - 7x - 4)

=3x^2(2x+1)(x-4)