Answer :

To express the given function into partial fractions, we'll break down the process into simpler steps.

Here's the expression we want to decompose:

[tex]\[
\frac{4x^4 + 4x^3 - 23x^2 - 4}{x^2 + x - 6}
\][/tex]

Step 1: Factor the Denominator

Before we start the decomposition, we need to factor the quadratic denominator [tex]\(x^2 + x - 6\)[/tex]. This can be factored into:

[tex]\[
(x - 2)(x + 3)
\][/tex]

Step 2: Setup the Partial Fraction Decomposition

With the denominator factored, the expression can be rewritten in partial fractions form:

[tex]\[
\frac{4x^4 + 4x^3 - 23x^2 - 4}{(x - 2)(x + 3)} = \frac{A}{x - 2} + \frac{B}{x + 3}
\][/tex]

Additionally, since the numerator has a degree higher than the denominator, we should expect a polynomial term. Let's express it as:

[tex]\[
4x^2 + 1 + \frac{A}{x - 2} + \frac{B}{x + 3}
\][/tex]

Step 3: Identify the Polynomial Part

Divide the polynomial in the numerator by the denominator polynomial to obtain the polynomial part, which in this solution is [tex]\(4x^2 + 1\)[/tex].

Step 4: Find the Residual Part

After extracting the polynomial part [tex]\(4x^2 + 1\)[/tex], we are left with a simpler expression to decompose:

[tex]\[
\frac{-1}{(x - 2)(x + 3)} = \frac{A}{x - 2} + \frac{B}{x + 3}
\][/tex]

To solve for constants [tex]\(A\)[/tex] and [tex]\(B\)[/tex], multiply through by the original denominator to clear the fractions:

[tex]\[
-1 = A(x + 3) + B(x - 2)
\][/tex]

Step 5: Solve for Constants

Now, expand and solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:

[tex]\[
-1 = Ax + 3A + Bx - 2B
\][/tex]

Combine like terms:

[tex]\[
-1 = (A + B)x + (3A - 2B)
\][/tex]

To solve, set the coefficients equal:

For [tex]\(x\)[/tex]:
[tex]\(A + B = 0\)[/tex]

For the constant term:
[tex]\(3A - 2B = -1\)[/tex]

Solve these equations:

1. From [tex]\(A + B = 0\)[/tex], you get [tex]\(B = -A\)[/tex].
2. Substitute into [tex]\(3A - 2(-A) = -1\)[/tex]:

[tex]\(3A + 2A = -1\)[/tex]

[tex]\(5A = -1\)[/tex]

[tex]\(A = -\frac{1}{5}\)[/tex]

3. Then [tex]\(B = -(-\frac{1}{5}) = \frac{1}{5}\)[/tex].

Final Expression in Partial Fractions

Putting it all together, the partial fraction decomposition is:

[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]

Therefore, the final decomposition of the given expression is:

[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]

This represents the function in terms of partial fractions.

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