Let's factor the polynomial [tex]\( 3x^4 - 2x^3 + 15x^2 - 10 \)[/tex] by grouping.
Given polynomial:
[tex]\[ 3x^4 - 2x^3 + 15x^2 - 10 \][/tex]
We can group the terms in pairs and factor each pair.
First, let's rewrite the polynomial in a way that groups terms effectively:
[tex]\[ (3x^4 - 2x^3) + (15x^2 - 10) \][/tex]
Now, factor out the common factor from each of these groups.
For the first group, [tex]\( 3x^4 - 2x^3 \)[/tex]:
[tex]\[ 3x^4 - 2x^3 = x^3(3x - 2) \][/tex]
For the second group, [tex]\( 15x^2 - 10 \)[/tex]:
[tex]\[ 15x^2 - 10 = 5(3x^2 - 2) \][/tex]
Now, combining these factored forms, we see that the polynomial can be grouped as:
[tex]\[ x^3(3x - 2) + 5(3x^2 - 2) \][/tex]
Notice that [tex]\( 3x^2 - 2 \)[/tex] appears as a common factor but isn't properly arranged together. Instead, we should check the original structure:
Taking a common intelligent grouping consideration:
Let’s consider:
[tex]\[ 3x^4 + 15x^2 - 2x^3 - 10 = (3x^4 + 15x^2) + (-2x^3 - 10) \][/tex]
[tex]\[ = 3x^2(x^2 + 5) - 2(x^2 + 5) \][/tex]
Factor out the common binomial [tex]\((x^2 + 5)\)[/tex]:
[tex]\[ (3x^2 + 2)(x^2 + 5) \][/tex]
Thus, the factored form is:
[tex]\[ (x^2 + 5)(3x^2 - 2) \][/tex]
So, the correct factored form of the polynomial is:
[tex]\[ \boxed{(x^2 + 5)(3x^2 - 2)} \][/tex]
