If $x+3$ is positive, then which one of the following must be positive?

Question

See: Nova GRE Math Bible. Page-$235$.

Problem#$09$
If $x+3$ is positive, then which one of the following must be positive?
(A) $x-3$
(B) $(x-3)(x-4)$
(C) $(x-3)(x+3)$
(D) $(x-3)(x+4)$
(E) $(x+3)(x+6)$

My attempted solution was as follows.

(A) $x-3 \Rightarrow x+3-6$. Since $x+3$ is positive, we have $-6$ left in our hand which is negative.
So, overall, (A) is negative.

(B) $(x-3)(x-4)$
$\Rightarrow (x+3-6)(x+3-7)$.
According to the previous logic, since we have $-6$ in the first term and $-7$ in the second term, the multiplication of them is positive.
So, overall, (B) is positive.

(C) $(x-3)(x+3)$
$\Rightarrow (x+3-6)(x+3)$.
According to the same previous logic, since we have $-6$ left in the first term,
(C) is overall negative.

(D) $(x-3)(x+4)$
$\Rightarrow (-6)(1)$.
So, (D) is negative.

(E) $(x+3)(x+6)\Rightarrow (3)$.
So, (E) is positive.

So, my answer is: (B), (E).
But, the given answer is (E)-only.

What is wrong with my explanation?

Answer 1

Here is a valid deduction for A):

Let $x = -1$. Then $x+3$ is positive, but $x-3$ is negative. So $x-3$ needn't be positive if $x + 3$ is positive.

Here is a valid deduction for B):

Let $x = 3$. Then $x+3$ is positive, but $x-3 = 0$. So $(x−4)(x-3) = 0$ So B needn't be positive if $x + 3$ is positive.

I'm sure you can do the rest.

Answer 2

$E$ is the only answer because $B$ is not always true.

For example if $x=3.5$ then $ x>-3 $ but $(x-3)(x-4)=-0.25 <0$