Which of the following equations in x is not a quadratic equation after simplification of both sides?

Introduction / Context:
This question asks you to identify which given equation is not quadratic once both sides are simplified. It tests your understanding of what defines a quadratic equation and your ability to expand and rearrange algebraic expressions correctly.

Given Data / Assumptions:

• Several equations in x are listed.
• We must simplify each and check whether the final form contains an x^2 term.
• A quadratic equation in one variable has the general form ax^2 + bx + c = 0 with a not equal to zero.

Concept / Approach:
Main ideas:

• Expand both sides of each equation.
• Bring all terms to one side to see the degree of the resulting polynomial.
• If the x^2 term cancels out completely, the equation is no longer quadratic.

Step-by-Step Solution:
Option a: 3x(x + 5) + 11 = 2x(x − 2) + 6. LHS: 3x^2 + 15x + 11. RHS: 2x^2 − 4x + 6. Subtract RHS: x^2 + 19x + 5 = 0, which is quadratic. Option b: 4x(x + 3) + 7 = 4x(x − 11) + 9. LHS: 4x^2 + 12x + 7. RHS: 4x^2 − 44x + 9. Subtract RHS: (4x^2 + 12x + 7) − (4x^2 − 44x + 9) = 56x − 2 = 0. This simplifies to 56x − 2 = 0, which is linear, not quadratic. Option c: x(x + 2) + 15 = x(2x − 5) + 11 gives −x^2 + 7x + 4 = 0, which is quadratic. Option d: 4x^2 − 6x + 9 = 0 is already quadratic. Option e: x^2 + 4x + 4 = 0 is also clearly quadratic.

Verification / Alternative check:
For option b, solving 56x − 2 = 0 gives a single solution x = 1/28. A genuine quadratic equation would have at most two roots and must involve an x^2 term. Since the x^2 terms cancel completely, the equation reduces to degree one, confirming that it is not quadratic.

Why Other Options Are Wrong:
Options a, c, d, and e all result in expressions containing x^2 after simplification. They match the general form ax^2 + bx + c = 0 with a not equal to zero. Therefore, each of them is a quadratic equation in x.

Common Pitfalls:
The main pitfall is performing expansions or subtractions incorrectly, leading to wrong cancellations of x^2 terms. Another mistake is to judge the equation by its initial appearance without actually simplifying both sides. Always expand, collect like terms, and then check the highest power of x.

Final Answer:
The equation that is not quadratic after simplification is 4x(x + 3) + 7 = 4x(x − 11) + 9.