We can start by simplifying the left side of the equation using the distributive property:
2x(3x+5)+3(3x+5) = (2x)(3x) + (2x)(5) + (3)(3x) + (3)(5)
= 6x^2 + 10x + 9x + 15
= 6x^2 + 19x + 15
Now we can compare this expression with the right side of the equation, which is a polynomial in x with unknown coefficients a, b, and c:
ax^2 + bx + c
Since the two sides are equal, their corresponding coefficients must be equal as well. This gives us a system of three equations in three unknowns:
a = 6 (the coefficient of x^2)
b = 19 (the coefficient of x)
c = 15 (the constant term)
Therefore, the solution to the equation 2x(3x+5)+3(3x+5)=ax^2+bx+c is:
2x(3x+5)+3(3x+5) = 6x^2 + 19x + 15