Final answer:
To factorize the expression 35m²ⁿ + 5mn - 30n using the middle term, we can split the middle term and then factor out the common binomial factor.
Step-by-step explanation:
To factorize the expression 35m²ⁿ + 5mn - 30n, we can use the middle term splitting method. First, we look for two numbers, let's call them a and b, such that a * b = 35 * -30 = -1050 and a + b = 5. We can solve this system of equations to find that the numbers are 35 and -30.
Now, we can split the middle term 5mn using these two numbers: 35m²ⁿ - 30mn + 35mn - 30n
Next, we can factor each pair of terms separately: 35m²ⁿ - 30mn + 35mn - 30n = 5m(7mⁿ - 6n) + 5n(7mⁿ - 6n)
Finally, we can factor out the common binomial factor of (7mⁿ - 6n): (7mⁿ - 6n)(5m + 5n)