Solve the Quadratic Equation: ax² + 5a + x = (3 + a)x² - (x + a)²

Solve the following equation:

$ax^2+5a+x=(3+a)x^2-(x+a)^2$

To solve the given equation $ax^2 + 5a + x = (3+a)x^2 - (x+a)^2$, we begin with expansion and simplification:

• Step 1: Expand the terms on the right side:
  $(3+a)x^2 - (x+a)^2 = (3+a)x^2 - (x^2 + 2ax + a^2) = (3+a)x^2 - x^2 - 2ax - a^2$
• Step 2: Simplify further:
  $(3+a)x^2 - x^2 - 2ax - a^2 = 2ax^2 - 2ax - a^2 + 3x^2$
• Step 3: Collect and equate coefficients from both sides:
  $0 = (2a + 2a - a)x^2 + (a - 5)x - a^2$
• Step 4: Set each type of coefficient separately to zero, assuming that the equation is valid for all $x$:
  - Coefficient of $x^2$: $a = 3$
  - Coefficient of $x$: $1 = 2a - 1$
  Solving these inequalities in terms of $a$ gives us the final inequality solution.

From the analysis, the solution is constrained by the inequalities derived from the simplification process. Hence, the answer is:
Thus, the solution to the problem is $-3.644 \ge a, -0.023 \le a$.