Divide using long division. State the quotient, $q(x)$, and the remainder, $r(x)$.
\[
\frac{4 x^{3}+10 x^{2}+5 x-21}{x+2}
\]
\[
\frac{4 x^{3}+10 x^{2}+5 x-21}{x+2}=\square+\frac{\square}{x+2}
\]
(Simplify your answers. Do not factor.)
Answer
Final Answer: The quotient is \(\boxed{4x^2 + 2x - 3}\) and the remainder is \(\boxed{-15}\).
Step 1 ：The problem is asking to perform polynomial long division. The dividend is \(4x^3 + 10x^2 + 5x - 21\) and the divisor is \(x + 2\). The quotient will be the result of the division and the remainder will be what is left over after the division.
Step 2 ：Perform the polynomial long division.
Step 3 ：The quotient, \(q(x)\), is \(4x^2 + 2x - 3\) and the remainder, \(r(x)\), is \(-15\).
Step 4 ：So, the division can be expressed as: \[\frac{4 x^{3}+10 x^{2}+5 x-21}{x+2}=4x^2 + 2x - 3+\frac{-15}{x+2}\]
Step 5 ：Final Answer: The quotient is \(\boxed{4x^2 + 2x - 3}\) and the remainder is \(\boxed{-15}\).
