Solving the Equation: (x+1)(2x)(3x+5)(x+2)=4x^2+1
This article will guide you through the process of solving the given equation. We'll expand the products, simplify the expression, and then solve for x.
Expanding the Products
First, we need to expand the products on the lefthand side of the equation:

(x+1)(2x): Using the distributive property (or FOIL method), we get: (x+1)(2x) = 2x  x² + 2  x = x² + x + 2

(3x+5)(x+2): Again, using the distributive property, we get: (3x+5)(x+2) = 3x² + 6x + 5x + 10 = 3x² + 11x + 10
Now, we can substitute these expanded expressions back into the original equation:
x² + x + 2  (3x² + 11x + 10) = 4x² + 1
Simplifying the Expression
Next, we simplify the equation by distributing the negative sign and combining like terms:
x² + x + 2  3x²  11x  10 = 4x² + 1
4x²  10x  8 = 4x² + 1
Solving for x
Now, we need to solve for x. Notice that the 4x² terms cancel out on both sides:
10x  8 = 1
Add 8 to both sides:
10x = 9
Divide both sides by 10:
x = 9/10
Therefore, the solution to the equation (x+1)(2x)(3x+5)(x+2)=4x^2+1 is x = 9/10.