Solving the Equation: (m4)(3m10) = 3(m2) + 18
This article will guide you through the steps of solving the equation (m4)(3m10) = 3(m2) + 18. We will use algebraic manipulations to isolate the variable m and find its value.
Step 1: Expand both sides of the equation
First, we need to expand the products on both sides of the equation.
 Left side: (m4)(3m10) = 3m²  10m  12m + 40 = 3m²  22m + 40
 Right side: 3(m2) + 18 = 3m  6 + 18 = 3m + 12
Now the equation becomes: 3m²  22m + 40 = 3m + 12
Step 2: Move all terms to one side
To solve for m, we need to have a quadratic equation (an equation with a term containing m²). To achieve this, subtract 3m + 12 from both sides:
3m²  22m + 40  (3m + 12) = 0
Simplifying the equation, we get: 3m²  25m + 28 = 0
Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation. We can factor it as follows:
(3m  4)(m  7) = 0
Step 4: Solve for m
For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:

3m  4 = 0
 Adding 4 to both sides: 3m = 4
 Dividing both sides by 3: m = 4/3

m  7 = 0
 Adding 7 to both sides: m = 7
Conclusion
The solutions to the equation (m4)(3m10) = 3(m2) + 18 are m = 4/3 and m = 7.