We are tasked with finding how many prime numbers $p$ make the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>+</mo><mn>7</mn><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3p^2 + p + 1 + 7p^2 + p + 1</annotation></semantics></math>3p2+p+1+7p2+p+1 divisible by $p$.

First, simplify the given expression:

Now, we want to determine for which prime numbers $p$, the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">10p^2 + 2p + 2</annotation></semantics></math>10p2+2p+2 is divisible by $p$. To do this, we need to check the divisibility condition:

By reducing the expression modulo $p$, we get:

Since <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>≡</mo><mn>0</mn><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p^2 \equiv 0 \pmod{p}</annotation></semantics></math>p2≡0(modp) and $p\equiv 0(\mathrm{mod}p)$, the expression simplifies to:

Therefore, the condition becomes:

This implies that $p$ must divide 2. The only prime number that divides 2 is $p=2$.

Thus, the only prime number that satisfies the divisibility condition is $p=2$.

Answer: There is 1 prime number, which is $p=2$. 

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