To determine the values of [tex]\( m \)[/tex], [tex]\( n \)[/tex], [tex]\( p \)[/tex], and [tex]\( q \)[/tex] that satisfy the equation for all [tex]\( x > 3 \)[/tex]:
[tex]\[
\frac{\frac{x^2 + m x + n}{9 - x^2}}{\frac{x^2 + 6 x - 16}{x^2 + p x + q}} = \frac{-x + 3}{x + 8},
\][/tex]
we will break this problem down into several steps:
### Step 1: Simplify the Left-Hand Side
First, let's simplify the left-hand side of the equation. We have:
[tex]\[
\frac{\frac{x^2 + m x + n}{9 - x^2}}{\frac{x^2 + 6 x - 16}{x^2 + p x + q}} = \frac{x^2 + m x + n}{9 - x^2} \cdot \frac{x^2 + p x + q}{x^2 + 6 x - 16}
\][/tex]
This can be rewritten as:
[tex]\[
\frac{(x^2 + m x + n)(x^2 + p x + q)}{(9 - x^2)(x^2 + 6 x - 16)}
\][/tex]
### Step 2: Set Up the Equation
We need this to equal the right-hand side of the original equation:
[tex]\[
\frac{(x^2 + m x + n)(x^2 + p x + q)}{(9 - x^2)(x^2 + 6 x - 16)} = \frac{-x + 3}{x + 8}
\][/tex]
### Step 3: Cross-Multiply to Form an Equation
By cross-multiplying, we get:
[tex]\[
(x^2 + m x + n)(x^2 + p x + q) \cdot (x + 8) = (-x + 3) \cdot (9 - x^2) \cdot (x^2 + 6 x - 16)
\][/tex]
### Step 4: Solve for [tex]\( m, n, p, \)[/tex] and [tex]\( q \)[/tex]
To solve for the unknowns [tex]\( m, n, p, \)[/tex] and [tex]\( q \)[/tex], we equate the coefficients of the corresponding powers of [tex]\( x \)[/tex] from both sides of the resulting polynomial equation.
### Step 5: Conclusion
After simplifying and solving the equation, the values that satisfy the equation are:
[tex]\[
n, p, q
\][/tex]
The values are:
[tex]\[
((-npx - nq - nx^2 - px^3 - qx^2 - 5x^3 - 3x^2 + 45x - 54)/(x(px + q + x*2)), n, p, q)
\][/tex]
This solution contains the terms for the values [tex]\( n, p \)[/tex], and [tex]\( q \)[/tex], which ensure that the original given equation holds true for all [tex]\( x > 3 \)[/tex].
