Question

If A, equals, p, squared, minus, 5, p, plus, 3A=p
2
−5p+3 and B, equals, minus, 2, p, minus, 1, commaB=−2p−1, find an expression that equals 2, A, plus, 2, B2A+2B in standard form.

Answer 1

To find the expression for \(2A + 2B\), we’ll first break down \(A\) and \(B\) as given:

1. \(A = p^2 - 5p + 3\)
2. \(B = -2p - 1\)

Next, we calculate \(2A\) and \(2B\):

\[
2A = 2(p^2 - 5p + 3) = 2p^2 - 10p + 6
\]

\[
2B = 2(-2p - 1) = -4p - 2
\]

Now we combine these results to find \(2A + 2B\):

\[
2A + 2B = (2p^2 - 10p + 6) + (-4p - 2)
\]

Now, we combine like terms:

1. The \(p^2\) term: \(2p^2\)
2. The \(p\) terms: \(-10p - 4p = -14p\)
3. The constant terms: \(6 - 2 = 4\)

Thus, the final expression \(2A + 2B\) simplifies to:

\[
2A + 2B = 2p^2 - 14p + 4
\]

This expression is already in standard form. Therefore, the final result is:

\[
\boxed{2p^2 - 14p + 4}
\]