Factor 3v^2 - 42v + 144

To factor the quadratic expression 3v^2 - 42v + 144, we need to find two numbers that multiply to give 144 and add to give -42.

The factors of 144 are:
1, 144
2, 72
3, 48
4, 36
6, 24
8, 18
9, 16
12, 12

The pair of factors that add to give -42 is 6 and 24.

Now we can rewrite the middle term -42v as -6v - 36v.

So, the expression can be factored as follows:
3v^2 - 42v + 144 = 3v^2 - 6v - 36v + 144

We can then factor by grouping:
3v^2 - 6v - 36v + 144 = (3v^2 - 6v) + (-36v + 144)
= 3v(v - 2) - 36(v - 4)

Finally, we factor out the common factors:
3v(v - 2) - 36(v - 4) = 3v(v - 2) - 36(v - 4)
= 3v(v - 2) - 36(v - 4)
= (v - 2)(3v - 36)

Therefore, the fully factored form of 3v^2 - 42v + 144 is (v - 2)(3v - 36).