To factor the expression [tex]\(6x^4 - 5x^2 + 12x^2 - 10\)[/tex] by grouping, follow these steps:
1. Combine like terms:
The expression has terms with [tex]\(x^2\)[/tex] that should be combined. So,
[tex]\[
6x^4 - 5x^2 + 12x^2 - 10 = 6x^4 + 7x^2 - 10
\][/tex]
2. Factor the resulting expression:
We need to factor [tex]\(6x^4 + 7x^2 - 10\)[/tex].
3. Identify the structure for factoring:
The expression can be viewed as a quadratic in terms of [tex]\(x^2\)[/tex]:
[tex]\[
6(x^2)^2 + 7(x^2) - 10
\][/tex]
Let [tex]\(u = x^2\)[/tex]. Then, the expression becomes:
[tex]\[
6u^2 + 7u - 10
\][/tex]
Now, factor [tex]\(6u^2 + 7u - 10\)[/tex].
4. Find factors that multiply to give the constant term and sum to give the middle coefficient:
We look for two numbers that multiply to [tex]\(6 \cdot (-10) = -60\)[/tex] and add to [tex]\(7\)[/tex].
These numbers are [tex]\(12\)[/tex] and [tex]\(-5\)[/tex].
5. Rewrite the middle term using these numbers:
[tex]\[
6u^2 + 12u - 5u - 10
\][/tex]
6. Group terms and factor each group:
[tex]\[
(6u^2 + 12u) + (-5u - 10)
\][/tex]
Factor out the common terms from each group:
[tex]\[
6u(u + 2) - 5(u + 2)
\][/tex]
7. Factor out the common binomial factor [tex]\(u + 2\)[/tex]:
[tex]\[
(6u - 5)(u + 2)
\][/tex]
8. Substitute back [tex]\(u = x^2\)[/tex]:
[tex]\[
(6x^2 - 5)(x^2 + 2)
\][/tex]
Thus, the factored expression is:
[tex]\[
(6x^2 - 5)(x^2 + 2)
\][/tex]
Therefore, the resulting expression is [tex]\(\boxed{\left(6 x^2-5\right)\left(x^2+2\right)}\)[/tex].
