Question

consider the equation, where f, g, and h are real numbers.
\\(\frac{2x + 1}{x - 3} + \frac{-3x + 2}{4x + 7} = \frac{fx^2 + gx + h}{(x - 3)(4x + 7)}\\)
what are the values of f, g, and h?

\\(\circ\\) \\(f = 11, g = 7,\\) and \\(h = -1\\)

\\(\circ\\) \\(f = 11, g = 29,\\) and \\(h = 1\\)

\\(\circ\\) \\(f = 5, g = 7,\\) and \\(h = -1\\)

\\(\circ\\) \\(f = 5, g = 29,\\) and \\(h = 1\\)

Explanation:

Step1: Combine left-hand fractions

$$\frac{(2x+1)(4x+7) + (-3x+2)(x-3)}{(x-3)(4x+7)} = \frac{Fx^2+Gx+H}{(x-3)(4x+7)}$$

Step2: Expand first numerator term

$$(2x+1)(4x+7) = 8x^2 + 14x + 4x + 7 = 8x^2 + 18x + 7$$

Step3: Expand second numerator term

$$(-3x+2)(x-3) = -3x^2 + 9x + 2x - 6 = -3x^2 + 11x - 6$$

Step4: Sum expanded terms

$$(8x^2 + 18x + 7) + (-3x^2 + 11x - 6) = 5x^2 + 29x + 1$$

Step5: Match coefficients to right-hand side

Since denominators are equal, equate numerator coefficients:
$F=5$, $G=29$, $H=1$

Answer:

F. $F = 5, G = 29$, and $H = 1$