The equation given in the image is:

$x^3 + 4x^2y + axy^2 + 3xy - bx^cy + 7xy^2 + dxy + y^2 = x^3 + y^2$

We need to find the values of $a + b + c + d$.

Step-by-step solution:

1. Simplify both sides of the equation:

   • The right-hand side is $x^3 + y^2$.
   • The left-hand side contains several terms involving powers of $x$ and $y$.

2. Eliminate matching terms:

   • The $x^3$ term appears on both sides of the equation, so it cancels out.
   • The $y^2$ term also appears on both sides, so it cancels out as well.

Thus, the equation simplifies to:

$4x^2y + axy^2 + 3xy - bx^cy + 7xy^2 + dxy = 0$

3. Group like terms:
   • Terms involving $xy$: $4x^2y$, $3xy$, $-bx^cy$, and $dxy$.
   • Terms involving $xy^2$: $axy^2$ and $7xy^2$.

So the equation can be rearranged as:

$4x^2y + (a + 7)xy^2 + (3 + d - bx^c)xy = 0$

4. Solve for the unknowns:

   • For the equation to hold for all values of $x$ and $y$, the coefficients of $xy$, $xy^2$, and $x^2y$ must be zero.

   • For the $xy^2$ terms: $a + 7 = 0 \quad \Rightarrow \quad a = -7$

   • For the $xy$ terms: $3 + d - bx^c = 0$ Since there is no other power of $x$ than $x^1$, $c = 1$, and we get: $3 + d - b = 0 \quad \Rightarrow \quad b = 3 + d$

   • For the $x^2y$ term: $4x^2y = 0$ There is no $x^2y$ term on the right-hand side, so this term must vanish, which implies no extra conditions here.

5. Conclusion: From the equations above:

   • $a = -7$
   • $b = 3 + d$
   • $c = 1$

The problem asks for $a + b + c + d$. Substitute the values:

$a + b + c + d = -7 + (3 + d) + 1 + d = -7 + 3 + 1 + 2d = -3 + 2d$

If we assume $d = 0$ (the simplest assumption to balance the terms), we get:

$a + b + c + d = -3$

Thus, the sum is:

$a + b + c + d = -3$

Would you like further details or have any questions about this?

Related Questions:

1. What if $d$ were not equal to zero?
2. How can we solve more complex polynomial equations?
3. Can this equation be solved using a system of equations approach?
4. How do we handle equations with multiple variables?
5. What is the role of powers in simplifying polynomial equations?

Tip:

When solving polynomial equations, always compare corresponding terms of the same degree to find relationships between the unknown coefficients.