<Question>
If $3x + 2(1 + x) = 17$, what is the value of $6x + 5$?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the value of 6x + 5 given the equation 3x + 2(1 + x) = 17. Hmm, where do I start?

First, maybe I should simplify the given equation step by step. Let me rewrite it to make sure I have it right: 3x plus 2 times (1 plus x) equals 17. Alright, so I need to solve for x first, right? Because once I know x, I can plug it into 6x + 5 and find the answer.

So, starting with the equation: 3x + 2(1 + x) = 17. Let me distribute the 2 into the parentheses.

<prompt_1.1> Simplify the equation 3x + 2(1 + x) = 17 by distributing the 2 into the parentheses. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
That should give me 3x + 2*1 + 2*x. Let me check that. Yeah, distributive property says a(b + c) = ab + ac. So, 2*(1 + x) is 2*1 + 2*x, which is 2 + 2x. So, substituting back into the equation, it becomes 3x + 2 + 2x = 17.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, combine like terms. The terms with x are 3x and 2x.
3x plus 2x is 5x. So, the equation simplifies to 5x + 2 = 17.

Next, I need to solve for x. Let me subtract 2 from both sides to isolate the term with x.

<prompt_2.1> Solve the simplified equation 5x + 2 = 17 for x by isolating the term with x. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
5x = 17 - 2. 17 minus 2 is 15, so 5x = 15. Then, divide both sides by 5 to get x by itself. So, x = 15 / 5, which is 3.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check if I did that correctly. Let me substitute x = 3 back into the original equation to verify.

<prompt_3.1> Verify the solution by substituting x = 3 back into the original equation 3x + 2(1 + x) = 17. </prompt_3.1>

Now, the question is asking for the value of 6x + 5. Since x is 3, let's compute that.
6 times 3 is 18, and 18 plus 5 is 23. So, 6x + 5 equals 23 when x is 3.

But hold on, maybe there's another way to approach this without solving for x first? Let me think. Sometimes, depending on the problem, you can manipulate the original equation to get the desired expression. Let's see if that's possible here.

<prompt_3.2> Explore an alternative method to find 6x + 5 without solving for x by manipulating the original equation. </prompt_3.2>

Alternatively, another way to see this is that 6x + 5 is like multiplying the original equation by 2 and then adjusting. Let me try that.

<prompt_3.3> Verify the result by multiplying the original equation by 2 and adjusting to find 6x + 5. </prompt_3.3>

But just to make sure I didn't make any mistakes in my steps, let me go through them again.

<prompt_3.4> Review the steps taken to solve for x and find 6x + 5 to ensure accuracy. </prompt_3.4>

Another thing to check is arithmetic. For example, when I distributed the 2, did I do that correctly? 2 times 1 is 2, and 2 times x is 2x. Yes.

<prompt_3.5> Verify the arithmetic and calculations in the solution process. </prompt_3.5>

Also, when combining like terms, 3x + 2x is 5x.
Yes, that's correct. Then 5x + 2 = 17. Subtract 2: 5x = 15. Divide by 5: x = 3. All steps look good.

Substituting back into 6x +5: 6*3=18, 18+5=23. Perfect.

Is there any chance I misread the original equation? Let me check again.

<prompt_3.6> Confirm the original equation and verify the solution against it. </prompt_3.6>
</Plan_3>

<Execution_3>
<execution_3.1>
The original equation was 3x + 2(1 + x) = 17. Plugging in x = 3: 3*3 + 2*(1 + 3) = 9 + 2*4 = 9 + 8 = 17. Yep, that adds up to 17, which matches the equation.
</execution_3.1>

<execution_3.2>
The original equation is 3x + 2(1 + x) = 17. Let's expand that again: 3x + 2 + 2x = 17, which is 5x + 2 = 17. If I solve for 5x, that's 17 - 2 = 15, so 5x = 15. Then, 6x would be (6/5)*(5x) = (6/5)*15. Let's see, 6/5 of 15 is 6*3 = 18. Then, adding 5 gives 18 + 5 = 23.
</execution_3.2>

<execution_3.3>
Original equation: 3x + 2(1 + x) = 17. If I multiply the entire equation by 2, would that help? Let's see. Wait, maybe not. Alternatively, notice that 6x is twice 3x. So, if I can express 6x +5 in terms of the original equation.

Starting with the original equation: 3x + 2(1 + x) = 17. Let's compute 3x first. If 3x = 17 - 2(1 + x). But maybe that's more complicated. Alternatively, solve for 3x. Let's see.

Wait, 3x + 2 + 2x = 17. So, 5x + 2 =17. So, 5x=15, x=3. Then 6x +5 is 6*3 +5=23. So, same result.

Alternatively, maybe express 6x +5 in terms of the original equation. Let's see. Let's note that 6x is 2*(3x). From the original equation, 3x = 17 - 2(1 + x) -2x? Wait, maybe this approach is convoluted.

Alternatively, once we have x=3, plug into 6x +5. Which is straightforward. So, as we did first.
</execution_3.3>

<execution_3.4>
Original equation: 3x + 2(1 + x) = 17. Distributed the 2: 3x + 2 + 2x = 17. Combined 3x + 2x to get 5x. So, 5x + 2 = 17. Subtract 2: 5x = 15. Divide by 5: x = 3. Then plug into 6x + 5: 6*3 +5. 6*3 is 18, 18+5 is 23.
</execution_3.4>

<execution_3.5>
Then adding that to 3x gives 5x. Then 5x +2=17. Subtract 2: 15. 15 divided by 5 is 3. Then 6*3=18, 18+5=23.
</execution_3.5>

<execution_3.6>
The user wrote: 3x + 2(1 + x) = 17. So, yes, that's correct. No parentheses missing or misread. So, all steps are accurate.

Alternatively, suppose I didn't solve for x, but tried to express 6x +5 in terms of the original equation. Let's see.

Original equation: 3x + 2(1 + x) =17. Let me expand that again: 3x + 2 + 2x =17, which is 5x +2=17. So, 5x=15, x=3. Then, 6x +5=6*3 +5=23.
</execution_3.6>
</Execution_3>

<Final_answer>
Therefore, the value of $6x + 5$ is $\boxed{23}$.
</Final_answer>
</think>