To subtract these radicals together their radicands must be the same:

4\( \sqrt{45} \) - 3\( \sqrt{5} \)
4\( \sqrt{9 \times 5} \) - 3\( \sqrt{5} \)
4\( \sqrt{3^2 \times 5} \) - 3\( \sqrt{5} \)
(4)(3)\( \sqrt{5} \) - 3\( \sqrt{5} \)
12\( \sqrt{5} \) - 3\( \sqrt{5} \)

Now that the radicands are identical, you can subtract them:

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{6!}{2!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{6 \times 5 \times 4 \times 3}{1} \)
\( 6 \times 5 \times 4 \times 3 \)
360

To multiply terms with radicals, multiply the coefficients and radicands separately:

4\( \sqrt{4} \) x 8\( \sqrt{8} \)
(4 x 8)\( \sqrt{4 \times 8} \)
32\( \sqrt{32} \)

Now we need to simplify the radical:

32\( \sqrt{32} \)
32\( \sqrt{2 \times 16} \)
32\( \sqrt{2 \times 4^2} \)
(32)(4)\( \sqrt{2} \)
128\( \sqrt{2} \)

First, solve for \( \left|z + 6\right| \):

\( \left|z + 6\right| \) + 8 = 9
\( \left|z + 6\right| \) = 9 - 8
\( \left|z + 6\right| \) = 1

The value inside the absolute value brackets can be either positive or negative so (z + 6) must equal + 1 or -1 for \( \left|z + 6\right| \) to equal 1:

So, z = -7 or z = -5.