To factor the polynomial [tex]\( 6x^4 - 5x^2 + 12x^2 - 10 \)[/tex] by grouping, let's follow these steps:
1. Combine like terms:
We first need to simplify the expression by combining the like terms [tex]\( -5x^2 \)[/tex] and [tex]\( 12x^2 \)[/tex]:
[tex]\[
6x^4 - 5x^2 + 12x^2 - 10 = 6x^4 + 7x^2 - 10
\][/tex]
2. Identify a method to factor:
To factor [tex]\( 6x^4 + 7x^2 - 10 \)[/tex], we look for ways to split and simplify the polynomial into factors.
3. Analyze the polynomial:
The polynomial [tex]\( 6x^4 + 7x^2 - 10 \)[/tex] is a quadratic in terms of [tex]\( x^2 \)[/tex]. We can use substitution to make it clearer. Let [tex]\( y = x^2 \)[/tex]. Substituting [tex]\( y \)[/tex] into the polynomial gives us:
[tex]\[
6y^2 + 7y - 10
\][/tex]
4. Factor the quadratic expression:
[tex]\( 6y^2 + 7y - 10 \)[/tex] can be factored by finding two numbers that multiply to [tex]\( 6 \times (-10) = -60 \)[/tex] and add to [tex]\( 7 \)[/tex]. These numbers are [tex]\( 12 \)[/tex] and [tex]\( -5 \)[/tex].
Rewrite [tex]\( 7y \)[/tex] as [tex]\( 12y - 5y \)[/tex]:
[tex]\[
6y^2 + 12y - 5y - 10
\][/tex]
Group the terms:
[tex]\[
(6y^2 + 12y) + (-5y - 10)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
6y(y + 2) - 5(y + 2)
\][/tex]
Now, factor out the common binomial factor:
[tex]\[
(y + 2)(6y - 5)
\][/tex]
5. Substitute back [tex]\( y = x^2 \)[/tex]:
Undo the substitution [tex]\( y = x^2 \)[/tex]:
[tex]\[
(x^2 + 2)(6x^2 - 5)
\][/tex]
So, the factored form of the polynomial [tex]\( 6x^4 + 7x^2 - 10 \)[/tex] is:
[tex]\[
(x^2 + 2)(6x^2 - 5)
\][/tex]
Therefore, the correct factorization is:
[tex]\[
\boxed{(6x^2 - 5)(x^2 + 2)}
\][/tex]
