Step 1: Understanding the Factor Theorem
The Factor Theorem states that if a polynomial \( f(x) \) has a factor \( (x - c) \), then \( f(c) = 0 \). This means that \( c \) is a root of the polynomial.

Step 2: Identifying Coefficients
For the polynomial \( 6x^2 + 17x + 5 \):
- \( a = 6 \)
- \( b = 17 \)
- \( c = 5 \)

Step 3: Finding Possible Rational Roots
Using the Rational Root Theorem, we can find potential rational roots by considering factors of \( c \) (5) over factors of \( a \) (6):
- Factors of 5: \( \pm 1, \pm 5 \)
- Factors of 6: \( \pm 1, \pm 2, \pm 3, \pm 6 \)
Potential rational roots are \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{5}{1}, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6} \).

Step 4: Testing Possible Roots
We can substitute each potential root into the polynomial to find a value of \( x \) that gives \( f(x) = 0 \).
- Testing \( x = \frac{1}{2} \):
\[
f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^2 + 17\left(\frac{1}{2}\right) + 5 = 6\left(\frac{1}{4}\right) + \frac{17}{2} + 5
\]
\[
= \frac{3}{2} + \frac{17}{2} + 5 = 0
\]
Thus, \( x = \frac{1}{2} \) is a root.

Step 5: Factoring the Polynomial
Since \( x - \frac{1}{2} \) is a factor, we can perform polynomial long division to factor \( 6x^2 + 17x + 5 \) as:
\[
6x^2 + 17x + 5 = (x - \frac{1}{2})(6x + 10)
\]
This can be simplified further as \( (2x + 1)(3x + 5) \).

Final Result
The factorization of \( 6x^2 + 17x + 5 \) is:
\[
(2x + 1)(3x + 5)
\]