For "predicted mean house price when holding area fixed at its mean" they're asking you to solve for Y-hat (predicted Y) when you plug in the mean value of area into the equation and add the value of the coefficient for "West".
𝑦𝑖=𝛽0+𝛽1×(PLUG IN MEAN VALUE OF AREA)+𝛽2×(1)+𝛽3×(0)+𝛽4×(0) [assuming you used Northeast as the base level and assuming "West" is coded 1 if West, 0 if not and the others are similarly coded].
Also, I would say that in order to use this model you would first want to fit a complete, second-order model with interactions of area and location if you have data to do so because the effect of area on price may be quadratic in area (diminishing returns, so to speak), and the effect of area may depend on the location (or effect of location may depend on area). You could then do a nested F-test on all of the interaction terms and retain them in the model if the test is significant. You could then do a t-test on the curvature (area-squared) term and retain if significant. There are other ways to approaching the actual retention of terms portion, though.
If this is a homework problem, and you have not covered interaction or squared terms in the models, then disregard those parts as they likely want you to fit a model like yours (but as another poster mentioned, with k-1 dummy variables for k locations).
Finally, the mean house price for West would be beta zero + beta 2 IF and only IF the area is equal to zero sq ft, which is nonsensical. So this tells you that a plausible (preferably in sample) value of area be chosen to plug in to get the mean value of price for West (they tell you to use average area).