Answer
She needs to score an $84$.
Work Step by Step
Recall, the equation for average is the sum of the values divided by the total number of values. Thus, we find:
$$ \frac{74+88+76+78+x}{5}=80\\ \frac{5\left(74+88+76+78+x\right)}{5}=80\cdot \:5\\ x+316=400\\ x=84$$
