Partial fraction decomposition of (x + 7)/((x - 2)(x + 1)) with detailed algebraic steps.

The problem involves expressing the rational function \(\frac{x+7}{(x-2)(x+1)}\) in partial fractions. Let's go through the solution step by step and explain each part.

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Step 1: Set up the partial fraction decomposition

The given rational function is:
\[
\frac{x+7}{(x-2)(x+1)}
\]
Since the denominator \((x-2)(x+1)\) consists of distinct linear factors, we can write:
\[
\frac{x+7}{(x-2)(x+1)} = \frac{A}{x-2} + \frac{B}{x+1}
\]
where \(A\) and \(B\) are constants to be determined.

Combining the right-hand side over a common denominator \((x-2)(x+1)\), we get:
\[
\frac{A}{x-2} + \frac{B}{x+1} = \frac{A(x+1) + B(x-2)}{(x-2)(x+1)}
\]
Equating the numerators, we have:
\[
x + 7 = A(x+1) + B(x-2)
\]

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Step 2: Expand and simplify the equation

Expand the right-hand side:
\[
A(x+1) + B(x-2) = Ax + A + Bx - 2B = (A + B)x + (A - 2B)
\]
Thus, the equation becomes:
\[
x + 7 = (A + B)x + (A - 2B)
\]

By comparing coefficients of \(x\) and the constant terms on both sides, we get two equations:
1. Coefficient of \(x\): \(A + B = 1\)
2. Constant term: \(A - 2B = 7\)

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Step 3: Solve the system of equations

We now solve the system of linear equations:
\[
\begin{cases}
A + B = 1 \\
A - 2B = 7
\end{cases}
\]

#### Eliminate \(A\):
Subtract the second equation from the first:
\[
(A + B) - (A - 2B) = 1 - 7
\]
\[
A + B - A + 2B = -6
\]
\[
3B = -6
\]
\[
B = -2
\]

#### Solve for \(A\):
Substitute \(B = -2\) into the first equation \(A + B = 1\):
\[
A + (-2) = 1
\]
\[
A - 2 = 1
\]
\[
A = 3
\]

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Step 4: Write the partial fraction decomposition

Now that we have \(A = 3\) and \(B = -2\), the partial fraction decomposition is:
\[
\frac{x+7}{(x-2)(x+1)} = \frac{3}{x-2} + \frac{-2}{x+1}
\]
or equivalently:
\[
\frac{x+7}{(x-2)(x+1)} = \frac{3}{x-2} - \frac{2}{x+1}
\]

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Step 5: Verify the solution

To verify, combine the partial fractions back into a single fraction:
\[
\frac{3}{x-2} - \frac{2}{x+1} = \frac{3(x+1) - 2(x-2)}{(x-2)(x+1)}
\]
Expand the numerator:
\[
3(x+1) - 2(x-2) = 3x + 3 - 2x + 4 = x + 7
\]
Thus, the combined fraction is:
\[
\frac{x+7}{(x-2)(x+1)}
\]
This matches the original expression, confirming that our solution is correct.

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Final Answer

\[
\boxed{\frac{3}{x-2} - \frac{2}{x+1}}
\]