To determine the factors of the polynomial [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] by grouping, follow these steps:
1. Group the terms: The given polynomial is [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex]. We want to group terms in pairs that can be factored easily.
[tex]\[ (x^3 + 5x^2) + (-6x - 30) \][/tex]
2. Factor out the common factors in each group:
- For the first group [tex]\( x^3 + 5x^2 \)[/tex], the common factor is [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 (x + 5) \][/tex]
- For the second group [tex]\( -6x - 30 \)[/tex], the common factor is [tex]\(-6\)[/tex]:
[tex]\[ -6(x + 5) \][/tex]
3. Write the expression after factoring out the common factors:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]
4. Factor out the common binomial factor [tex]\((x + 5)\)[/tex]:
[tex]\[ (x^2 - 6)(x + 5) \][/tex]
Therefore, the correct way to factor [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] by grouping is:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]
This corresponds to the option:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]
So, the correct choice is:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]
