Which polynomial is prime? $$ \begin{array}{l}x^{4}+2 x^{2} \ 25 y^{2}+49 \ x^{4}-144 \ y^{3}+64 \ \end{array} $$

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We check each polynomial for factorability over the integers.

1. For 
$$
x^4+2x^2:
$$
we notice a common factor:
$$
x^4+2x^2=x^2(x^2+2),
$$
so it is composite.

2. For 
$$
25y^2+49:
$$
this is a quadratic in $$ y $$. A quadratic polynomial is irreducible (prime) if it cannot be factored into linear factors with integer (or rational) coefficients. In this case, any factorization would have the form
$$
(ay+b)(cy+d)
$$
with $$ ac=25 $$ and $$ bd=49 $$. However, no choice of integers $$ a,b,c,d $$ yields the middle term necessary to reconstruct the original polynomial. Hence, $$25y^2+49$$ is irreducible (prime).

3. For 
$$
x^4-144:
$$
we recognize a difference of squares:
$$
x^4-144=(x^2)^2-12^2=(x^2-12)(x^2+12),
$$
so it is composite.

4. For 
$$
y^3+64:
$$
this is a sum of cubes:
$$
y^3+64=y^3+4^3=(y+4)(y^2-4y+16),
$$
so it is composite.

Thus, the only prime polynomial in the list is 
$$
25y^2+49.
$$
$$25y^2+49$$ is the prime polynomial.

Which polynomial is prime? \( \begin{array}{l}x^{4}+2 x^{2} \\ 25 y^{2}+49 \\ x^{4}-144 \\ y^{3}+64 \\ \end{array} \)

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