Questions: Factor completely. -2v^2+11v+21

Solution Steps

To factor the quadratic expression \(-2v^2 + 11v + 21\), we need to find two numbers that multiply to the product of the leading coefficient and the constant term (i.e., \(-2 \times 21 = -42\)) and add up to the middle coefficient (i.e., \(11\)). Once these numbers are found, we can use them to split the middle term and factor by grouping.

We start with the quadratic expression: \[ -2v^2 + 11v + 21 \]

To factor the expression, we can rewrite it as: \[ -(2v^2 - 11v - 21) \] Next, we find two numbers that multiply to \(-42\) (the product of \(-2\) and \(21\)) and add to \(11\). The numbers \(14\) and \(-3\) satisfy these conditions.

We can split the middle term \(11v\) using \(14v\) and \(-3v\): \[ -2v^2 + 14v - 3v + 21 \]

Now, we group the terms: \[ (-2v^2 + 14v) + (-3v + 21) \] Factoring each group gives us: \[ -2v(v - 7) - 3(v - 7) \] This can be factored further as: \[ -(v - 7)(2v + 3) \]

Final Answer