Solve -4(x²+5) = (-x+7)(4x-9)+5: Complete Quadratic Solution

To solve this equation, we'll follow these steps:

• Step 1: Expand and simplify the right-hand side.
• Step 2: Set the equation to zero by moving all terms to one side.
• Step 3: Simplify to obtain a standard quadratic equation.
• Step 4: Use the quadratic formula to find the possible solutions for $x$.

Now, let's work through each step:

Step 1:
Expand the right-hand side:
$(-x + 7)(4x - 9) = -x(4x) - x(-9) + 7(4x) - 7(9)$
= $-4x^2 + 9x + 28x - 63$
Considering both sides: $-4(x^2 + 5) = -4x^2 + 9x + 28x - 63 + 5$.

Step 2:
Simplify further by calculating:
$-4x^2 - 20 = -4x^2 + 37x - 58$.

Step 3:
Move all terms to one side to achieve zero on the right-hand side:
$-4x^2 - 20 + 4x^2 - 37x + 58 = 0$
Simplifying, we get: $37x + 38 = 0$.

Step 4:
Since the $x^2$ terms cancel, it's actually a linear equation:
$37x = -38$.
Solving for $x$, we divide both sides by 37:
$x = \frac{-38}{37} = -1\frac{1}{37}$.

Therefore, the solution to the problem is $x = 1\frac{1}{37}$.