Decompose the polynomial into linear factors: (x2−7x+6)(x2+3x−18)

To decompose the polynomial \((x^2 − 7x + 6)(x^2 + 3x − 18)\) into linear factors, we first need to factor each quadratic expression separately.

1. Factor \(x^2 - 7x + 6\):

   • We are looking for two numbers that multiply to \(6\) (the constant term) and add to \(-7\) (the coefficient of \(x\)).
   • The numbers are \(-1\) and \(-6\).
   • Thus, \(x^2 - 7x + 6 = (x - 1)(x - 6)\).

2. Factor \(x^2 + 3x - 18\):

   • We are looking for two numbers that multiply to \(-18\) and add to \(3\).
   • The numbers are \(6\) and \(-3\).
   • Thus, \(x^2 + 3x - 18 = (x + 6)(x - 3)\).

Now we can combine the factored forms: \[ (x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 1)(x - 6)(x + 6)(x - 3) \]

Putting all together, the final linear factored form of the polynomial is: \[ (x - 1)(x - 6)(x + 6)(x - 3) \]

However, the answer options provided do not exactly match this expression. The closest option appears to be:

(x−6)(x−1)(x+6)(x−3)

So, the correct option is: \[ \text{(x−6)(x−1)(x+6)(x−3)} \]