To determine the other linear factor of the trinomial \(2x^2 + 13x + 6\) given that one of its linear factors is \(x + 6\), we can follow these steps:
1. Identify the given trinomial and linear factor:
The trinomial is \(2x^2 + 13x + 6\).
The given linear factor is \(x + 6\).
2. Set up the polynomial division:
We need to divide the trinomial \(2x^2 + 13x + 6\) by the linear factor \(x + 6\).
3. Perform the polynomial division:
To find the other factor, we closely divide \(2x^2 + 13x + 6\) by \(x + 6\).
- First, divide the leading term of the trinomial \(2x^2\) by the leading term of the linear factor \(x\), which gives \(2x\).
- Multiply \(2x\) by \(x + 6\), resulting in \(2x^2 + 12x\).
- Subtract \(2x^2 + 12x\) from \(2x^2 + 13x + 6\) to get \(x + 6\).
- Divide \(x\) by \(x\), resulting in \(1\).
- Multiply \(1\) by \(x + 6\), resulting in \(x + 6\).
- Subtract \(x + 6\) from \(x + 6\), resulting in zero.
4. Result of the division:
The quotient from the division is \(2x + 1\). Since the remainder is zero, \(2x + 1\) is the other factor.
So, the trinomial factorizes as:
[tex]\[2x^2 + 13x + 6 = (x + 6)(2x + 1)\][/tex]
Thus, the other linear factor is \(2x + 1\).
Therefore, the correct choice from the provided options is:
[tex]\[ \boxed{2x + 1} \][/tex]
