Use synthetic division and the Remainder Theorem to evaluate \( P(c) \) \[ P(x)=3 x^{2}+15 x+2, \quad c=-4 \] \( P(-4)= \)

To evaluate \( P(-4) \) using synthetic division, we first set up the synthetic division for the polynomial \( P(x) = 3x^2 + 15x + 2 \) with \( c = -4 \). 1. **Set up the synthetic division**: We will use the coefficients of \( P(x) \), which are \( 3, 15, 2 \). 2. **Write the synthetic division**: \[ \begin{array}{r|rrr} -4 & 3 & 15 & 2 \\ & & -12 & -12 \\ \hline & 3 & 3 & -10 \\ \end{array} \] 3. **Perform the synthetic division**: - Bring down the leading coefficient \( 3 \). - Multiply \( -4 \) by \( 3 \) to get \( -12 \) and add it to \( 15 \) to get \( 3 \). - Multiply \( -4 \) by \( 3 \) to get \( -12 \) and add it to \( 2 \) to get \( -10 \). 4. **Result of synthetic division**: The bottom row gives us the coefficients of the quotient polynomial and the remainder. The quotient is \( 3x + 3 \) and the remainder is \( -10 \). 5. **Using the Remainder Theorem**: According to the Remainder Theorem, \( P(c) \) is equal to the remainder when \( P(x) \) is divided by \( x - c \). Therefore, \( P(-4) = -10 \). Thus, the value of \( P(-4) \) is \[ \boxed{-10}. \]