Simplifying the Algebraic Expression

The problem asks us to simplify the algebraic expression:

$ 4\left((\frac{3}{4})x^2 - 26x + 17\right) - 3(x^2 + 4x - 14) $

To simplify this, we will use the distributive property and combine like terms.

Step 1: Distribute the Factor 4

First, multiply the factor 4 by each term inside the first set of parentheses:

• $4 \times (\frac{3}{4})x^2 = \frac{12}{4}x^2 = 3x^2$
• $4 \times (-26x) = -104x$
• $4 \times 17 = 68$

After distributing the 4, the first part of the expression becomes:

$ 3x^2 - 104x + 68 $

Step 2: Distribute the Factor -3

Next, multiply the factor -3 by each term inside the second set of parentheses:

• $-3 \times x^2 = -3x^2$
• $-3 \times (4x) = -12x$
• $-3 \times (-14) = 42$

After distributing the -3, the second part of the expression becomes:

$ -3x^2 - 12x + 42 $

Step 3: Combine the Expanded Expressions

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

$ (3x^2 - 104x + 68) + (-3x^2 - 12x + 42) $

Step 4: Group Like Terms

Group the terms that have the same variable and exponent together:

• Combine the $x^2$ terms: $3x^2 + (-3x^2)$
• Combine the $x$ terms: $-104x + (-12x)$
• Combine the constant terms: $68 + 42$

Rewrite the expression with grouped terms:

$ (3x^2 - 3x^2) + (-104x - 12x) + (68 + 42) $

Step 5: Simplify by Performing Operations

Perform the addition and subtraction for each group of like terms:

• $3x^2 - 3x^2 = 0x^2 = 0$
• $-104x - 12x = -116x$
• $68 + 42 = 110$

Combine the simplified terms:

$ 0 - 116x + 110 $

Final Simplified Expression

The fully simplified expression is:

$ -116x + 110 $