To factor the trinomial \(x^2 - x - 30\), we aim to rewrite it as the product of two binomials. The general form of such a trinomial factorization is:
[tex]\[
(x + a)(x + b)
\][/tex]
We need to find suitable values of \(a\) and \(b\) such that the equation holds true. Specifically, we are looking for \(a\) and \(b\) which satisfy the following conditions:
1. \(a \cdot b = -30\) (The constant term of the product)
2. \(a + b = -1\) (The coefficient of the middle term)
Considering these criteria, we brainstorm pairs of factors of \(-30\) and check if their sum equals \(-1\):
For example:
- \(a = 5\) and \(b = -6\):
- Their product: \(5 \cdot (-6) = -30\)
- Their sum: \(5 + (-6) = -1\)
Thus, we find that the suitable values for \(a\) and \(b\) are 5 and -6, respectively. Therefore, the factors of the trinomial \(x^2 - x - 30\) can be expressed as:
[tex]\[
(x + 5)(x - 6)
\][/tex]
Now, we compare these factors with the provided options:
- Option A: \(x + 3\)
- Option B: \(x - 6\)
- Option C: \(x + 10\)
- Option D: \(x + 5\)
From the pairs \((x + 5)\) and \((x - 6)\):
The correct binomials that are factors of the trinomial \(x^2 - x - 30\) are:
[tex]\[
\boxed{B \text{ and } D}
\][/tex]
