Simplifying the Algebraic Expression

The problem asks us to simplify the given algebraic expression:

$4\left((\frac{6}{4})x^2 - 25x + 12\right) - 6(x^2 + 5x - 13)$

To simplify this, we need to follow the order of operations (PEMDAS/BODMAS) and apply the distributive property.

Step-by-Step Simplification

• Step 1: Simplify fractions and distribute the first term.

  First, simplify the fraction $\frac{6}{4}$ to $\frac{3}{2}$. The expression inside the first parenthesis becomes $\frac{3}{2}x^2 - 25x + 12$. Now, distribute the 4:

  $4 \times (\frac{3}{2}x^2 - 25x + 12) = (4 \times \frac{3}{2})x^2 - (4 \times 25)x + (4 \times 12)$

  $= 6x^2 - 100x + 48$

• Step 2: Distribute the second term.

  Next, distribute the -6 into the second parenthesis $(x^2 + 5x - 13)$:

  $-6 \times (x^2 + 5x - 13) = (-6 \times x^2) + (-6 \times 5x) + (-6 \times -13)$

  $= -6x^2 - 30x + 78$

• Step 3: Combine the results.

  Now, add the results from Step 1 and Step 2:

  $(6x^2 - 100x + 48) + (-6x^2 - 30x + 78)$

• Step 4: Group and combine like terms.

  Group the terms with the same power of x:

  $ (6x^2 - 6x^2) + (-100x - 30x) + (48 + 78) $

  Combine the coefficients:

  $ 0x^2 - 130x + 126 $

• Step 5: Final simplified expression.

  The simplified expression is:

  $ -130x + 126 $

Conclusion

After performing the distribution and combining like terms, the simplified form of the expression $4\left((\frac{6}{4})x^2 - 25x + 12\right) - 6(x^2 + 5x - 13)$ is $-130x + 126$.