Questions: Factor (6 x^4-5 x^2+12 x^2-10) by grouping. What is the resulting expression? ((6 x+5)(x^2-2)) ((6 x-5)(x^2+2)) ((6 x^2+5)(x^2-2)) ((6 x^2-5)(x^2+2))

Transcript text: Factor $6 x^{4}-5 x^{2}+12 x^{2}-10$ by grouping. What is the resulting expression? $(6 x+5)\left(x^{2}-2\right)$ $(6 x-5)\left(x^{2}+2\right)$ $\left(6 x^{2}+5\right)\left(x^{2}-2\right)$ $\left(6 x^{2}-5\right)\left(x^{2}+2\right)$

Solution

Solution Steps

To factor the polynomial $6x^4 - 5x^2 + 12x^2 - 10$ by grouping, we can follow these steps:

Combine like terms to simplify the polynomial.
Group the terms in pairs.
Factor out the greatest common factor (GCF) from each group.
Look for a common binomial factor in the resulting expression.

Step 1: Combine Like Terms

We start with the polynomial $6x^4 - 5x^2 + 12x^2 - 10$. Combining the like terms, we have: \[ 6x^4 + 7x^2 - 10 \]

Step 2: Group the Terms

Next, we group the polynomial into two pairs: \[ (6x^4 + 7x^2) + (-10) \]

Step 3: Factor Each Group

Now, we factor out the greatest common factor from each group: \[ x^2(6x^2 + 7) - 10 \] However, we can also rearrange and factor it differently: \[ (6x^2 - 5)(x^2 + 2) \]