Let's use algebra to solve this problem.

Let A be the number of apple seeds and O be the number of orange seeds that Sue planted.

According to the given information:

1. A = 4O (Sue planted 4 times as many apple seeds as orange seeds).
2. 15% of the apple seeds grew into trees, so 0.15A apple trees grew.
3. 10% of the orange seeds grew into trees, so 0.10O orange trees grew.
4. The total number of trees is 420: A + O = 420.

Now, we can set up a system of equations based on the above information:

From equation 1, we have: A = 4O

From equations 2 and 3, we know the total number of trees: 0.15A + 0.10O = 420

Now, substitute the value of A from equation 1 into equation 2: 0.15(4O) + 0.10O = 420

Simplify the equation: 0.60O + 0.10O = 420

Combine like terms: 0.70O = 420

Now, solve for O (the number of orange seeds): O = 420 / 0.70 O = 600

So, Sue planted 600 orange seeds.