To factorise the quadratic expression \(6x² + 17x + 5\) using the Factor Theorem, follow these steps:

Step 1: Identify the coefficients
- The quadratic can be expressed as \(ax^2 + bx + c\), where:
- \(a = 6\)
- \(b = 17\)
- \(c = 5\)

Step 2: Find possible rational roots
- According to the Rational Root Theorem, the possible rational roots are of the form \(±\frac{p}{q}\), where \(p\) is a factor of \(c\) (5) and \(q\) is a factor of \(a\) (6).
- Factors of 5: \(±1, ±5\)
- Factors of 6: \(±1, ±2, ±3, ±6\)
- Possible rational roots: \(±1, ±\frac{1}{2}, ±\frac{1}{3}, ±\frac{1}{6}, ±5, ±\frac{5}{2}, ±\frac{5}{3}, ±\frac{5}{6}\)

Step 3: Test possible roots
- Substitute these values into the polynomial to find roots. For instance, testing \(x = 1\):
\[
6(1)^2 + 17(1) + 5 = 6 + 17 + 5 = 28 \quad (\text{not a root})
\]
- Testing \(x = \frac{1}{2}\):
\[
6\left(\frac{1}{2}\right)^2 + 17\left(\frac{1}{2}\right) + 5 = 6\left(\frac{1}{4}\right) + \frac{17}{2} + 5 = \frac{3}{2} + \frac{17}{2} + \frac{10}{2} = 0 \quad (\text{is a root})
\]

Step 4: Factor the polynomial
- Since \(x = \frac{1}{2}\) is a root, we can express the polynomial as \((x - \frac{1}{2})(\text{quadratic})\).
- To find the quadratic factor, perform synthetic division or polynomial long division.

Step 5: Write the final factorised form
- After performing the division, we find:
\[
6x^2 + 17x + 5 = (2x + 1)(3x + 5)
\]
This is the factorised form of the polynomial.