Read the following passage and answer the given questions. Two algebraic expressions are given as (I) $6x^2 + 29x + P$ and (II) $3x + 1$. (I) is divided by (II) leaving $(2x + 9)$ as quotient and 19 as remainder. What is the value of largest root of equation $6x^2 + 29x + P = 0$?

### Concept & Quadratic Equation Roots To find the roots of a quadratic equation, we can use factorization by splitting the middle term. $$ ax^2 + bx + c = 0 $$ ### Step-by-Step Solution * From the previous problem, we found that $P = 28$. * Substitute $P$ into the equation: $6x^2 + 29x + 28 = 0$. * To factorize, we need two numbers that multiply to $a \times c$ ($6 \times 28 = 168$) and add to $b$ ($29$). * These numbers are 21 and 8 ($21 + 8 = 29$ and $21 \times 8 = 168$). * Split the middle term: $6x^2 + 21x + 8x + 28 = 0$. * Factor by grouping: $3x(2x + 7) + 4(2x + 7) = 0$. * This gives $(3x + 4)(2x + 7) = 0$. * Setting each factor to zero yields the roots: $x = -\frac{4}{3}$ and $x = -\frac{7}{2}$. * Convert to mixed fractions/decimals to compare: $x = -1\frac{1}{3}$ and $x = -3.5$. * Since $-1.33$ is greater than $-3.5$, the largest root is $-1\frac{1}{3}$. ### Exam Strategy & Shortcut Once you have the roots as $-\frac{21}{6}$ and $-\frac{8}{6}$, you can immediately see that $-\frac{8}{6}$ (which simplifies to $-\frac{4}{3}$) is a smaller negative number and thus the larger value. There's no need to convert fully to decimals if you recognize relative fractional sizes. ### Common Pitfall Choosing $-\frac{7}{2}$ (or $-3.5$) instead of $-1.33$ because $3.5$ is a larger magnitude. Remember that on the negative side of the number line, numbers closer to zero have a larger mathematical value. ### Final Answer Therefore, the correct answer is **$-1\frac{1}{3}$**.