Simplifying the Algebraic Expression Step-by-Step

The problem requires simplifying the following algebraic expression:

To achieve this, we need to perform two main operations:

• Distribute the constant factors ($3$ and $-7$) into their respective parentheses.
• Combine the resulting like terms.

Distributing Constants in the Expression

First, distribute the $3$ into the terms inside the first parenthesis:

• $3 \times \left(\frac{7}{3}\right)x^2 = \frac{3 \times 7}{3}x^2 = 7x^2$
• $3 \times (-25x) = -75x$
• $3 \times 12 = 36$

The first part of the expression becomes: $7x^2 - 75x + 36$.

Next, distribute the $-7$ into the terms inside the second parenthesis:

• $-7 \times x^2 = -7x^2$
• $-7 \times (9x) = -63x$
• $-7 \times (-13) = +91$

The second part of the expression becomes: $-7x^2 - 63x + 91$.

Combining Like Terms

Now, combine the simplified parts of the expression:

Expression = $(7x^2 - 75x + 36) + (-7x^2 - 63x + 91)$

Group the terms with the same power of $x$ together:

• x² terms: $7x^2$ and $-7x^2$
• x terms: $-75x$ and $-63x$
• Constant terms: $36$ and $91$

Combine the coefficients for each group:

• For the $x^2$ terms: $7x^2 - 7x^2 = (7 - 7)x^2 = 0x^2 = 0$
• For the $x$ terms: $-75x - 63x = (-75 - 63)x = -138x$
• For the constant terms: $36 + 91 = 127$

Finally, sum the results of the combined terms:

The simplified expression is $-138x + 127$.