Note: This may be intended as a simple exercise of polynomial subtraction. Just combine like terms and see what you get. We can save the important technical considerations for later, but in the grand scheme of things, there is a whole lot more going on here than a subtraction problem. If it were my assignment, I'd be a whole lot more careful about the problem design.

This is a very tricky problem. You MUST decide what things mean. You may not just assume that these formulae are valid everywhere.

1) We probably should start with x >= 0, since you cannot ship fewer than zero (0) tons.

2) By similar considerations, we cannot ship more than 374 tons by train or 541 tons by truck. If we can, then someone will have to start paying us to use them. This is not reasonable.

3) Noting that trains quit at 374, but trucks continue to 541, there is NOT ANY solution to the problem, as stated, for 374 < x < 542. Don't even try to write one. It doesn't exist.

Having said that, simply subtract. \(\displaystyle (-0.09x^{2} + 34x - 100) - (-0.04x^{2} + 22x - 175) = ??\). Be very careful with the order of operations and the distributive property and combine like terms. Make sure you agree that this is applicable for only 0 <= x <= 374. It doesn't mean anything for any other value of x.