To determine the base of the triangle, we can use the formula for the area of a triangle:
Area = (base * height) / 2
Given that the area of the triangle is 2x^2 + 5x + 3, and the height is 4x + 6, we can substitute these values into the formula:
2x^2 + 5x + 3 = (base * (4x + 6)) / 2
To simplify the equation, multiply both sides by 2:
4x^2 + 10x + 6 = base * (4x + 6)
Now, distribute the base to both terms on the right side of the equation:
4x^2 + 10x + 6 = 4x * base + 6 * base
At this point, we have an equation with a quadratic term (x^2), a linear term (x), and a constant term (6). To solve for the base, we need to isolate the base term on one side of the equation. Rearranging the equation:
4x^2 + 10x + 6 - 4x * base = 6 * base
Now, let's factor out the base on the left side of the equation:
4x^2 + 10x + 6 - (4x * base) = 6 * base
4x^2 + 10x + 6 - 4x * base - 6 * base = 0
-4x * base - 6 * base = -4x^2 - 10x - 6
Factoring out the common term (base):
base * (-4x - 6) = -4x^2 - 10x - 6
Divide both sides by (-4x - 6) to solve for the base:
base = (-4x^2 - 10x - 6) / (-4x - 6)
Now, we have determined the base of the triangle in terms of x.