To prove that a=b=c when two equations have two common roots, let's analyze the given equations in detail.

Equation 1: ax^2 + bx + c = 0
Equation 2: x^3 + 3x^2 + 3x + 2 = 0

1. Find the common roots:
Let's assume that the two common roots are α and β.

2. Substitute the common roots in both equations:
Equation 1: aα^2 + bα + c = 0
Equation 2: α^3 + 3α^2 + 3α + 2 = 0

Equation 1: aβ^2 + bβ + c = 0
Equation 2: β^3 + 3β^2 + 3β + 2 = 0

3. Equate the two equations using the common roots:
By equating Equation 1 and Equation 2, we have:

aα^2 + bα + c = α^3 + 3α^2 + 3α + 2
aβ^2 + bβ + c = β^3 + 3β^2 + 3β + 2

4. Simplify the equations:
Subtracting Equation 1 from Equation 2, we get:

a(β^2 - α^2) + b(β - α) = β^3 - α^3 + 3(β^2 - α^2) + 3(β - α)

5. Factorize the difference of cubes:
Using the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2), we can rewrite the equation as:

a(β - α)(β^2 + αβ + α^2) + b(β - α) = (β - α)[(β^2 + αβ + α^2) + 3(β + α)]

6. Divide both sides by (β - α):
(β - α)[a(β^2 + αβ + α^2) + b] = (β - α)[(β^2 + αβ + α^2) + 3(β + α)]

7. Cancel out (β - α) from both sides:
a(β^2 + αβ + α^2) + b = (β^2 + αβ + α^2) + 3(β + α)

8. Rearrange the equation:
(a - 1)(β^2 + αβ + α^2) + (b - 3)(β + α) = 0

9. Since α and β are common roots, (β + α) and (β^2 + αβ + α^2) will be equal to zero.

10. Therefore, we have:
(a - 1)(0) + (b - 3)(0) = 0
0 + 0 = 0

11. Simplify the equation:
0 = 0

12. This equation is always true, regardless of the values of a, b, and c. Thus, we can conclude that a=b=c.

In summary, by substituting the common roots into the given equations and simplifying the expressions, we can prove that a=b=c

of that cubic eqn one root is -2..and rest 2 are imaginary...as imaginry roots occur in pair so that mst also b roots of ax2+bx+c ...so ax2+bx+c is equal to x2+x+1..