Factor. 4d^2 + 12d + 5 ____

Identify a, b, c: Identify a, b, and c in the quadratic expression 4d^2 + 12d + 5 by comparing it with the standard form ax^2 + bx + c. a = 4 b = 12 c = 5Find Multiplying Numbers: Find two numbers that multiply to a*c (which is 4*5=20) and add up to b (which is 12). We need to find two numbers that satisfy these conditions.Check Factors: After checking possible pairs of factors of 20, we find that 2 and 10 multiply to 20 and add up to 12. So the two numbers we are looking for are 2 and 10. 2 * 10 = 20 2 + 10 = 12Rewrite Middle Term: Rewrite the middle term (12d) using the two numbers we found (2 and 10) to split it into two terms. 4d^2 + 12d + 5 becomes 4d^2 + 2d + 10d + 5.Factor by Grouping: Factor by grouping. Group the first two terms and the last two terms. (4d^2 + 2d) + (10d + 5)Factor out Common Factor: Factor out the greatest common factor from each group. 2d(2d + 1) + 5(2d + 1)Since both groups contain the common factor (2d + 1), factor this out. The factored form of the expression is (2d + 1)(2d + 5).

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Factor. $\newline$ $4d^2 + 12d + 5$

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Algebra 1

Factor quadratics with other leading coefficients

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Q. Factor.

\newline

4d^2 + 12d + 5

Identify $a$ , $b$ , $c$ : Identify $a$ , $b$ , and $c$ in the quadratic expression $4d^2 + 12d + 5$ by comparing it with the standard form $ax^2 + bx + c$ .
$a = 4$
$b = 12$
$b$ $0$
Find Multiplying Numbers: Find two numbers that multiply to $a*c$ (which is $4*5=20$ ) and add up to $b$ (which is $12$ ). $\newline$ We need to find two numbers that satisfy these conditions.
Check Factors: After checking possible pairs of factors of $20$ , we find that $2$ and $10$ multiply to $20$ and add up to $12$ . $\newline$ So the two numbers we are looking for are $2$ and $10$ . $\newline$ $2 \times 10 = 20$ $\newline$ $2 + 10 = 12$
Rewrite Middle Term: Rewrite the middle term ( $12d$ ) using the two numbers we found ( $2$ and $10$ ) to split it into two terms. $\newline$ $4d^2 + 12d + 5$ becomes $4d^2 + 2d + 10d + 5$ .
Factor by Grouping: Factor by grouping. Group the first two terms and the last two terms. $\newline$ $(4d^2 + 2d) + (10d + 5)$
Factor out Common Factor: Factor out the greatest common factor from each group. $2d(2d + 1) + 5(2d + 1)$
Factor out Common Factor: Factor out the greatest common factor from each group. $\newline$ $2d(2d + 1) + 5(2d + 1)$ Since both groups contain the common factor $(2d + 1)$ , factor this out. $\newline$ The factored form of the expression is $(2d + 1)(2d + 5)$ .