To determine a factor of the polynomial \(x^3 + 5x^2 + 6x\), we can start by factoring out the common term in the expression.
The given polynomial can be factored as follows:
\[ x^3 + 5x^2 + 6x = x(x^2 + 5x + 6) \]
Now we need to factor the quadratic \(x^2 + 5x + 6\):
Looking for two numbers that multiply to \(6\) (the constant term) and add to \(5\) (the coefficient of the linear term):
The pairs of factors of \(6\) are:
- \(1\) and \(6\)
- \(2\) and \(3\)
The pair that adds up to \(5\) is \(2\) and \(3\). Thus, we can factor \(x^2 + 5x + 6\) as follows:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Putting it all together, we can express the original polynomial as:
\[ x^3 + 5x^2 + 6x = x(x + 2)(x + 3) \]
From this factorization, we can see that the linear factors are \(x\), \(x + 2\), and \(x + 3\).
Now let's check the options given:
- Option #1: \(x - 2\) (This is not a factor)
- Option #2: \(x - 3\) (This is not a factor)
- Option #3: \(x^2 + 5x + 6\) (This is a factor but not linear)
Since \(x\), \(x + 2\), and \(x + 3\) are linear factors, and we don't have one listed among the options, we find that none of the provided options are correct linear factors of the polynomial \(x^3 + 5x^2 + 6x\).
However, \(x^2 + 5x + 6\) is indeed a factor, but it is not linear. Based on the request for a linear factor, there isn't one in the options provided. If the question was asking for any factor, then option #3 would be the best fit.