The expression in the image is:

$\frac{11x^2 - 4}{(x^2 + 2)(x^2 + 3)}$

We want to resolve this expression into partial fractions. Given that the denominator consists of two irreducible quadratic factors, the partial fraction decomposition will have the form:

$\frac{11x^2 - 4}{(x^2 + 2)(x^2 + 3)} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{x^2 + 3}$

Step 1: Combine the fractions on the right-hand side

To combine the fractions, get a common denominator:

$\frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{x^2 + 3} = \frac{(Ax + B)(x^2 + 3) + (Cx + D)(x^2 + 2)}{(x^2 + 2)(x^2 + 3)}$

Step 2: Set the numerators equal

Since the denominators are the same, equate the numerators:

$11x^2 - 4 = (Ax + B)(x^2 + 3) + (Cx + D)(x^2 + 2)$

Step 3: Expand and collect like terms

Expanding the right-hand side:

$(Ax + B)(x^2 + 3) = Ax^3 + 3Ax + Bx^2 + 3B$ $(Cx + D)(x^2 + 2) = Cx^3 + 2Cx + Dx^2 + 2D$

Combining these:

$Ax^3 + Bx^2 + 3Ax + 3B + Cx^3 + Dx^2 + 2Cx + 2D$ $= (A + C)x^3 + (B + D)x^2 + (3A + 2C)x + (3B + 2D)$

Step 4: Match coefficients

Now, match the coefficients of corresponding powers of $x$:

For $x^3$ terms: $A + C = 0$

For $x^2$ terms: $B + D = 11$

For $x$ terms: $3A + 2C = 0$

For the constant term: $3B + 2D = -4$

Step 5: Solve the system of equations

From $A + C = 0$, we have: $C = -A$

Substituting $C = -A$ into $3A + 2C = 0$, we get: $3A + 2(-A) = 3A - 2A = A = 0$

Since $A = 0$, $C = -A = 0$.

Now, using $B + D = 11$ and $3B + 2D = -4$:

Substitute $D = 11 - B$ into $3B + 2D = -4$: $3B + 2(11 - B) = -4$ $3B + 22 - 2B = -4$ $B + 22 = -4$ $B = -26$

Then, using $D = 11 - B$: $D = 11 - (-26) = 11 + 26 = 37$

Step 6: Write the partial fraction decomposition

The partial fraction decomposition is:

$\frac{11x^2 - 4}{(x^2 + 2)(x^2 + 3)} = \frac{-26}{x^2 + 2} + \frac{37}{x^2 + 3}$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you handle partial fractions when the numerator is a polynomial of higher degree?
2. Can you apply partial fraction decomposition to integrals?
3. What is the role of irreducible quadratic factors in partial fractions?
4. How can you check if your partial fraction decomposition is correct?
5. What is the