Let the full price of each rosebush be denoted by \( x \). According to the problem, each rosebush is marked down by $12. Therefore, the sale price of each rosebush is:
\[ x - 12 \]
Clare purchased 4 rosebushes, and she paid a total of $96 for them. Thus, we can set up the equation based on the total cost:
\[ 4(x - 12) = 96 \]
Now, we will simplify the equation:
\[ 4x - 48 = 96 \]
Next, we can add 48 to both sides of the equation to isolate the term with \( x \):
\[ 4x = 96 + 48 \] \[ 4x = 144 \]
Now, we divide both sides by 4 to solve for \( x \):
\[ x = \frac{144}{4} = 36 \]
Thus, the full price of each rosebush is:
\[ \boxed{36} \]
Finally, we can verify the solution. If the full price of each rosebush is $36, then the sale price is:
\[ 36 - 12 = 24 \]
The total cost for 4 rosebushes would be:
\[ 4 \times 24 = 96 \]
Since this matches Clare's total payment, our solution is confirmed. The full price of each rosebush is indeed \( \boxed{36} \).