b4) \( 4 p^{2}+7 p q-2 q^{2} \)

To factor the quadratic expression \( 4p^{2} + 7pq - 2q^{2} \), follow these steps: ### Step 1: Set Up the Factoring Framework Assume the expression can be factored into the form: \[ ( ap + bq )( cp + dq ) \] where \( a, b, c, \) and \( d \) are constants to be determined. ### Step 2: Determine Coefficients For the expression \( 4p^{2} + 7pq - 2q^{2} \), we need: \[ a \cdot c = 4 \quad \text{(coefficient of } p^2 \text{)} \] \[ b \cdot d = -2 \quad \text{(constant term)} \] \[ a \cdot d + b \cdot c = 7 \quad \text{(coefficient of } pq \text{)} \] Let's try possible integer pairs for \( (a, c) \) and \( (b, d) \): - \( (a, c) = (4, 1) \) - To satisfy \( b \cdot d = -2 \), possible pairs are \( (1, -2) \) or \( (-1, 2) \) ### Step 3: Test the Combinations **First Combination:** \[ (a, c) = (4, 1) \quad \text{and} \quad (b, d) = (1, -2) \] \[ a \cdot d + b \cdot c = 4 \cdot (-2) + 1 \cdot 1 = -8 + 1 = -7 \quad (\text{Does not match } 7) \] **Second Combination:** \[ (a, c) = (4, 1) \quad \text{and} \quad (b, d) = (-1, 2) \] \[ a \cdot d + b \cdot c = 4 \cdot 2 + (-1) \cdot 1 = 8 - 1 = 7 \quad (\text{Matches}) \] ### Step 4: Write the Factored Form Using the second combination: \[ 4p^{2} + 7pq - 2q^{2} = (4p - q)(p + 2q) \] ### Verification Multiply the factors to verify: \[ (4p - q)(p + 2q) = 4p \cdot p + 4p \cdot 2q - q \cdot p - q \cdot 2q = 4p^{2} + 8pq - pq - 2q^{2} = 4p^{2} + 7pq - 2q^{2} \] The multiplication confirms that the factoring is correct. **Final Answer:** \[ 4p^{2} + 7pq - 2q^{2} = (4p - q)(p + 2q) \]