To expand and simplify the expression \((x+6)(5-x)\) and see if it equals \(x^2 - 36\), we can start by using the distributive property (also known as the FOIL method for binomials).

1. Expand \((x + 6)(5 - x)\):
\[
(x + 6)(5 - x) = x \cdot 5 + x \cdot (-x) + 6 \cdot 5 + 6 \cdot (-x)
\]
\[
= 5x - x^2 + 30 - 6x
\]
\[
= -x^2 - x + 30
\]

Now we have:
\[
(x + 6)(5 - x) = -x^2 - x + 30
\]

2. Next, we set this equal to \(x^2 - 36\):
\[
-x^2 - x + 30 = x^2 - 36
\]

3. Now, we will rearrange this equation:
\[
-x^2 - x + 30 - x^2 + 36 = 0
\]
\[
-2x^2 - x + 66 = 0
\]

4. To simplify, multiply the entire equation by -1:
\[
2x^2 + x - 66 = 0
\]

Now we have a quadratic equation. You can use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) with \(a = 2\), \(b = 1\), and \(c = -66\).

Calculating the discriminant:
\[
b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot (-66) = 1 + 528 = 529
\]

Now solve for \(x\):
\[
x = \frac{{-1 \pm \sqrt{529}}}{2 \cdot 2} = \frac{{-1 \pm 23}}{4}
\]

Calculating the two potential solutions:
1. \(x = \frac{22}{4} = 5.5\)
2. \(x = \frac{-24}{4} = -6\)

Thus, the solutions to the equation \((x + 6)(5 - x) = x^2 - 36\) are:
\[
x = 5.5 \quad \text{and} \quad x = -6
\]