Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor \( f \) over the real numbers. \[ T(x)=4 x^{4}+9 x^{3}+22 x^{2}+45 x+10 \] Find the real zeros of f . Select the correct choice below and fili in any answer boxes in your choice. A. \( x=-2,-\frac{1}{4} \) (Simplity your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression Use a comma to separate answers as needed ) B. There are no real zeros. Use the real zeros to factor f . \[ f(x)=\square \] (Simplify your answer. Type your answer in foctored form. Type an exact answer, using radicais as needed. Use integers or fractions for any rational numbers in the expression)

To apply the Rational Zeros Theorem to the polynomial function \( T(x) = 4x^4 + 9x^3 + 22x^2 + 45x + 10 \), we look for possible rational zeros using the factors of the constant term (10) divided by the factors of the leading coefficient (4). This yields possible rational zeros of \( \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{4}, \pm \frac{5}{4}, \pm \frac{10}{4} \). After testing these values, we find that \( x = -2 \) and \( x = -\frac{1}{4} \) are indeed real zeros. Now, we can factor \( T(x) \) using these zeros as follows: \[ T(x) = 4(x + 2)\left( x + \frac{1}{4} \right)(Ax^2 + Bx + C) \] Upon performing polynomial long division or synthetic division, we can work out the remaining quadratic which can also be factored or solved. Finally, we can express \( T(x) \) fully factored as: \[ T(x) = (x + 2)(x + \frac{1}{4})(4x^2 + 13x + 10) \] So, the final answers are: A. \( x = -2, -\frac{1}{4} \) And \[ T(x) = (x + 2)(4x^2 + 13x + 10) \]