Questions: The middle term of the trinomial has been rewritten. Now factor by grouping. 20 r^2 + 31 r s + 12 s^2 = 20 r^2 + 15 r s + 16 r s + 12 s^2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 20 r^2 + 15 r s + 16 r s + 12 s^2 = B. The trinomial is prime.

The middle term of the trinomial has been rewritten. Now factor by grouping. 20 r^2 + 31 r s + 12 s^2 = 20 r^2 + 15 r s + 16 r s + 12 s^2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 20 r^2 + 15 r s + 16 r s + 12 s^2 = B. The trinomial is prime.
Transcript text: The middle term of the trinomial has been rewritten. Now factor by grouping. \[ 20 r^{2}+31 r s+12 s^{2}=20 r^{2}+15 r s+16 r s+12 s^{2} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $20 r^{2}+15 r s+16 r s+12 s^{2}=$ $\square$ B. The trinomial is prime.
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Solution

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Solution Steps

To factor the given expression by grouping, we first split the middle term into two terms that allow us to group the expression into two binomials. Then, we factor out the greatest common factor from each group and look for a common binomial factor.

Step 1: Rewrite the Expression

The given expression is: \[ 20r^2 + 31rs + 12s^2 \] We rewrite the middle term \(31rs\) as \(15rs + 16rs\) to facilitate factoring by grouping: \[ 20r^2 + 15rs + 16rs + 12s^2 \]

Step 2: Group the Terms

Group the terms into two pairs: \[ (20r^2 + 15rs) + (16rs + 12s^2) \]

Step 3: Factor Each Group

Factor out the greatest common factor from each group:

  • From \(20r^2 + 15rs\), factor out \(5r\): \[ 5r(4r + 3s) \]
  • From \(16rs + 12s^2\), factor out \(4s\): \[ 4s(4r + 3s) \]
Step 4: Factor the Common Binomial

Notice that both groups contain the common binomial factor \((4r + 3s)\). Factor this out: \[ (4r + 3s)(5r + 4s) \]

Final Answer

The factored form of the expression is: \[ \boxed{(4r + 3s)(5r + 4s)} \] Thus, the answer is A.

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