(4) \( \frac{x^{2}(x+2)-4(x+2)}{(x+2)^{2}} \) 7) \( 120^{5} y+3 x y \) 3) \( \frac{4 x^{2}-1}{3 x^{2}+10 x+3} \div \frac{6 x^{2}+5 x+1}{4 x^{2}+11} \times \frac{9 x^{2}+6 x+1}{8 x^{2}-6 x+} \)

Sure, let's simplify each of the given algebraic expressions step by step. --- ### **Problem 4** Simplify the expression: \[ \frac{x^{2}(x + 2) - 4(x + 2)}{(x + 2)^{2}} \] **Step 1: Factor the Numerator** Notice that \((x + 2)\) is a common factor in the numerator: \[ x^{2}(x + 2) - 4(x + 2) = (x + 2)(x^{2} - 4) \] **Step 2: Simplify the Expression** Now, the expression becomes: \[ \frac{(x + 2)(x^{2} - 4)}{(x + 2)^{2}} = \frac{x^{2} - 4}{x + 2} \] **Step 3: Factor Further if Possible** Factor \(x^{2} - 4\) as a difference of squares: \[ x^{2} - 4 = (x - 2)(x + 2) \] So, \[ \frac{(x - 2)(x + 2)}{x + 2} = x - 2 \quad \text{(for } x \neq -2\text{)} \] **Final Simplified Form:** \[ x - 2 \quad \text{where } x \neq -2 \] --- ### **Problem 7** Simplify the expression: \[ 120^{5} y + 3xy \] **Step 1: Factor Out the Common Term** Both terms have a common factor of \(y\): \[ 120^{5} y + 3xy = y(120^{5} + 3x) \] **Step 2: (Optional) Evaluate \(120^{5}\) if Needed** If you need to express \(120^{5}\) numerically: \[ 120^{5} = 120 \times 120 \times 120 \times 120 \times 120 = 24,\!883,\!200,\!000 \] So, \[ y(24,\!883,\!200,\!000 + 3x) \] **Final Simplified Form:** \[ y(120^{5} + 3x) \quad \text{or} \quad y(24,\!883,\!200,\!000 + 3x) \] --- ### **Problem 3** Simplify the expression: \[ \frac{4x^{2} - 1}{3x^{2} + 10x + 3} \div \frac{6x^{2} + 5x + 1}{4x^{2} + 11} \times \frac{9x^{2} + 6x + 1}{8x^{2} - 6x + 1} \] **Note:** It seems there was a typo in the denominator of the last fraction. Assuming it's \(8x^{2} - 6x + 1\). **Step 1: Factor Each Polynomial** - **Numerator of the first fraction:** \[ 4x^{2} - 1 = (2x - 1)(2x + 1) \] - **Denominator of the first fraction:** \[ 3x^{2} + 10x + 3 = (3x + 1)(x + 3) \] - **Numerator of the second fraction (divisor):** \[ 6x^{2} + 5x + 1 = (2x + 1)(3x + 1) \] - **Denominator of the second fraction:** \[ 4x^{2} + 11 \quad (\text{Cannot be factored further}) \] - **Numerator of the third fraction:** \[ 9x^{2} + 6x + 1 = (3x + 1)^2 \] - **Denominator of the third fraction:** \[ 8x^{2} - 6x + 1 = (2x - 1)(4x - 1) \] **Step 2: Rewrite the Entire Expression with Factored Forms** \[ \frac{(2x - 1)(2x + 1)}{(3x + 1)(x + 3)} \div \frac{(2x + 1)(3x + 1)}{4x^{2} + 11} \times \frac{(3x + 1)^2}{(2x - 1)(4x - 1)} \] **Step 3: Change Division to Multiplication by Reciprocal** \[ \frac{(2x - 1)(2x + 1)}{(3x + 1)(x + 3)} \times \frac{4x^{2} + 11}{(2x + 1)(3x + 1)} \times \frac{(3x + 1)^2}{(2x - 1)(4x - 1)} \] **Step 4: Cancel Out Common Factors** - \(2x - 1\) cancels with \(2x - 1\) - \(2x + 1\) cancels with \(2x + 1\) - \((3x + 1)^2\) cancels with \((3x + 1)(3x + 1)\) **Remaining Terms:** \[ \frac{4x^{2} + 11}{(x + 3)(4x - 1)} \] **Final Simplified Form:** \[ \frac{4x^{2} + 11}{(x + 3)(4x - 1)} \] --- If there's a different intended form or if the last denominator in **Problem 3** was different, please let me know so I can adjust the solution accordingly!