Simplifying the Algebraic Expression

The problem asks us to simplify the algebraic expression: $3\left((\frac{6}{3})x^2 - 21x + 18\right) - 6(x^2 + 5x - 18)$ We need to carefully apply the order of operations and the distributive property to combine like terms.

Step 1: Simplify Inside the First Parenthesis

First, let's simplify the term inside the first parenthesis. We calculate $\frac{6}{3}$: $\frac{6}{3} = 2$ So the expression becomes: $3(2x^2 - 21x + 18) - 6(x^2 + 5x - 18)$

Step 2: Apply the Distributive Property (First Term)

Next, we distribute the $3$ to each term inside the first parenthesis: $3 \times (2x^2) = 6x^2$ $3 \times (-21x) = -63x$ $3 \times 18 = 54$ The first part of the expression simplifies to $6x^2 - 63x + 54$.

Step 3: Apply the Distributive Property (Second Term)

Now, we distribute the $-6$ to each term inside the second parenthesis: $-6 \times (x^2) = -6x^2$ $-6 \times (5x) = -30x$ $-6 \times (-18) = 108$ The second part of the expression simplifies to $-6x^2 - 30x + 108$.

Step 4: Combine the Simplified Parts

Now we combine the results from Step 2 and Step 3: $(6x^2 - 63x + 54) + (-6x^2 - 30x + 108)$ To simplify, we group and combine like terms.

Combine x² Terms

We have $6x^2$ from the first part and $-6x^2$ from the second part: $6x^2 - 6x^2 = 0x^2 = 0$

Combine x Terms

We have $-63x$ from the first part and $-30x$ from the second part: $-63x - 30x = -93x$

Combine Constant Terms

We have $54$ from the first part and $108$ from the second part: $54 + 108 = 162$

Final Simplified Expression

Putting the combined terms together, we get the final simplified expression: $0 - 93x + 162 = -93x + 162$