To rewrite the equation \( d^2 + 3x = 6x + 10 \) in standard quadratic form, let's first rearrange it. We'll move all terms to one side of the equation:
\[ d^2 + 3x - 6x - 10 = 0 \]
This simplifies to:
\[ d^2 - 3x - 10 = 0 \]
Now, we'll factor this equation. We can treat this as a quadratic in terms of \( x \):
\[ d^2 = 3x + 10 \]
To rewrite this as a standard form quadratic equation in \( x \), we would typically set it to zero:
\[ -3x + d^2 - 10 = 0 \]
Rearranging gives us:
\[ 3x = d^2 - 10 \Rightarrow 3x - d^2 + 10 = 0 \]
Reorganizing shows that we can isolate \( x \) leading to:
\[ x = \frac{d^2 + 10}{3} \]
Now, to find the integer values of \( x \) and whether they fit with the provided selections (2, 5), (-2, -5), (2, -5), (-2, 5), we need to understand which values of \( d \) would result in those \( x \) values.
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When \( x = 2 \): \[ 3(2) = d^2 + 10 \implies 6 = d^2 + 10 \implies d^2 = -4 \quad \text{(not valid)} \]
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When \( x = 5 \): \[ 3(5) = d^2 + 10 \implies 15 = d^2 + 10 \implies d^2 = 5 \quad \text{(valid, } d = \sqrt{5} \text{)} \]
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When \( x = -2 \): \[ 3(-2) = d^2 + 10 \implies -6 = d^2 + 10 \implies d^2 = -16 \quad \text{(not valid)} \]
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When \( x = -5 \): \[ 3(-5) = d^2 + 10 \implies -15 = d^2 + 10 \implies d^2 = -25 \quad \text{(not valid)} \]
Thus, the only valid result for integer \( x \) from this scenario comes from \( x = 5 \). None of the other conditions yield feasible matches.
Hence, there are no valid values for \( x \) from the provided options. However, if it were necessary to examine potential implications, values leading up to results for other \( d \) variations, we realize \( x = 5 \) yields solutions. Therefore, based on these exploration contexts, the provided options are limited, suggesting focus on the alternatives with regard to \( d \).
Regarding basic identification:
The correct values seem to be none of the above options hold since all scenarios lead to unusable \( d^2 \), while counting for integers solidifying necessary investigations on the alternative confirmations should coincide into a definitive value under the generated simplifications extended here.