4. Expand and simplity by callecting like terms aा \( (x+1)(x+2) \) b) \( (y+2)\left(y^{2}-2 y-1\right) \) द7 \( (3 m-2)(m-4) \) d) \( (2 x-1)\left(x^{2}-2 x-1\right) \) (7) \( -(x-5)^{2} \)

Let's expand and simplify each expression by collecting like terms. --- ### a) \((x + 1)(x + 2)\) **Step 1: Apply the distributive property (FOIL method).** \[ \begin{align*} (x + 1)(x + 2) &= x \cdot x + x \cdot 2 + 1 \cdot x + 1 \cdot 2 \\ &= x^2 + 2x + x + 2 \end{align*} \] **Step 2: Combine like terms.** \[ x^2 + 2x + x + 2 = x^2 + 3x + 2 \] **Final Answer:** \[ x^2 + 3x + 2 \] --- ### b) \((y + 2)(y^{2} - 2y - 1)\) **Step 1: Distribute each term in the first polynomial to each term in the second polynomial.** \[ \begin{align*} (y + 2)(y^{2} - 2y - 1) &= y \cdot y^2 + y \cdot (-2y) + y \cdot (-1) + 2 \cdot y^2 + 2 \cdot (-2y) + 2 \cdot (-1) \\ &= y^3 - 2y^2 - y + 2y^2 - 4y - 2 \end{align*} \] **Step 2: Combine like terms.** \[ y^3 - 2y^2 + 2y^2 - y - 4y - 2 = y^3 - 5y - 2 \] **Final Answer:** \[ y^3 - 5y - 2 \] --- ### c) \((3m - 2)(m - 4)\) **Step 1: Apply the distributive property.** \[ \begin{align*} (3m - 2)(m - 4) &= 3m \cdot m + 3m \cdot (-4) - 2 \cdot m - 2 \cdot (-4) \\ &= 3m^2 - 12m - 2m + 8 \end{align*} \] **Step 2: Combine like terms.** \[ 3m^2 - 12m - 2m + 8 = 3m^2 - 14m + 8 \] **Final Answer:** \[ 3m^2 - 14m + 8 \] --- ### d) \((2x - 1)(x^{2} - 2x - 1)\) **Step 1: Distribute each term in the first polynomial to each term in the second polynomial.** \[ \begin{align*} (2x - 1)(x^{2} - 2x - 1) &= 2x \cdot x^2 + 2x \cdot (-2x) + 2x \cdot (-1) - 1 \cdot x^2 - 1 \cdot (-2x) - 1 \cdot (-1) \\ &= 2x^3 - 4x^2 - 2x - x^2 + 2x + 1 \end{align*} \] **Step 2: Combine like terms.** \[ 2x^3 - 4x^2 - x^2 - 2x + 2x + 1 = 2x^3 - 5x^2 + 1 \] **Final Answer:** \[ 2x^3 - 5x^2 + 1 \] --- ### e) \(- (x - 5)^{2}\) **Step 1: Expand the squared binomial.** \[ (x - 5)^2 = (x)^2 - 2 \cdot x \cdot 5 + (5)^2 = x^2 - 10x + 25 \] **Step 2: Apply the negative sign to each term.** \[ - (x^2 - 10x + 25) = -x^2 + 10x - 25 \] **Final Answer:** \[ - x^2 + 10x - 25 \] --- Each expression has been expanded and simplified by collecting like terms.