Factor Trinomials Show work for all problems algebraically, using factor by grouping \[ \begin{array}{l}\text { 8. } 2 r^{2}+27 r+70 \\ \text { 10. } 7 a^{2}+53 a+28\end{array} \]

Sure! Let's factor each trinomial step-by-step using **factor by grouping**. --- ### **Problem 8:** Factor \( 2r^{2} + 27r + 70 \) **Step 1: Identify the coefficients** \[ a = 2, \quad b = 27, \quad c = 70 \] **Step 2: Compute \( a \times c \)** \[ a \times c = 2 \times 70 = 140 \] **Step 3: Find two numbers that multiply to \( 140 \) and add up to \( 27 \)** We need two numbers, \( m \) and \( n \), such that: \[ m \times n = 140 \quad \text{and} \quad m + n = 27 \] After checking factor pairs of \( 140 \): \[ 7 \times 20 = 140 \quad \text{and} \quad 7 + 20 = 27 \] So, \( m = 7 \) and \( n = 20 \). **Step 4: Rewrite the middle term using \( m \) and \( n \)** \[ 2r^{2} + 7r + 20r + 70 \] **Step 5: Group the terms** \[ (2r^{2} + 7r) + (20r + 70) \] **Step 6: Factor out the greatest common factor (GCF) from each group** \[ r(2r + 7) + 10(2r + 7) \] **Step 7: Factor out the common binomial factor \( (2r + 7) \)** \[ (r + 10)(2r + 7) \] **Final Factored Form:** \[ 2r^{2} + 27r + 70 = (r + 10)(2r + 7) \] --- ### **Problem 10:** Factor \( 7a^{2} + 53a + 28 \) **Step 1: Identify the coefficients** \[ a = 7, \quad b = 53, \quad c = 28 \] **Step 2: Compute \( a \times c \)** \[ a \times c = 7 \times 28 = 196 \] **Step 3: Find two numbers that multiply to \( 196 \) and add up to \( 53 \)** We need two numbers, \( m \) and \( n \), such that: \[ m \times n = 196 \quad \text{and} \quad m + n = 53 \] After checking factor pairs of \( 196 \): \[ 4 \times 49 = 196 \quad \text{and} \quad 4 + 49 = 53 \] So, \( m = 4 \) and \( n = 49 \). **Step 4: Rewrite the middle term using \( m \) and \( n \)** \[ 7a^{2} + 4a + 49a + 28 \] **Step 5: Group the terms** \[ (7a^{2} + 4a) + (49a + 28) \] **Step 6: Factor out the greatest common factor (GCF) from each group** \[ a(7a + 4) + 7(7a + 4) \] **Step 7: Factor out the common binomial factor \( (7a + 4) \)** \[ (a + 7)(7a + 4) \] **Final Factored Form:** \[ 7a^{2} + 53a + 28 = (a + 7)(7a + 4) \] --- **Summary of Factored Forms:** 1. \( 2r^{2} + 27r + 70 = (r + 10)(2r + 7) \) 2. \( 7a^{2} + 53a + 28 = (a + 7)(7a + 4) \)